Comportamiento asintótico en tiempo de ecuaciones en derivadas parciales no locales

Programa de Doctorado “Matem´aticas” PhD Dissertation Comportamiento asint´otico en tiempo de ecuaciones en derivadas parciales no locales Long-time behaviour of nonlocal partial di↵erential equations Author Marta Herrera Cobos Supervisor Prof. Dr. Tom´as Caraballo Garrido Co-supervisor Prof. Dr. Pedro Mar´ın Rubio March 31, 2016 This work has been supported by Ministerio de Econom´ıa y Competitividad (Spain), grant MTM2011–22411. Programa de FPI del Ministerio de Econom´ıa y Competitividad, reference BES–2012–053398. A mis padres, por su apoyo incondicional A Antonio Jes´us, mi amigo, mi confidente, mi compa˜nero para siempre Agradecimientos Esta tesis doctoral no habr´ıa sido posible sin el apoyo y el cari˜no de todas las personas que me han acompa˜nado durante estos a˜nos. Me gustar´ıa empezar dando las gracias al profesor Jos´e Real por toda su ayuda al inicio de esta etapa. Aunque desafortunadamente no me ha podido acompa˜nar durante la elaboraci´on de mi tesis doctoral, ´el fue quien me ense˜n´o a caminar en este mundo, dej´andome en las manos de mis directores de tesis, los profesores Tom´as Caraballo Garrido y Pedro Mar´ın Rubio, gracias a los cuales este proyecto se ha hecho realidad y a los que tanto les debo. Gracias por todo los que me hab´eis ense˜nado, por vuestro esfuerzo y dedicaci´on, y sobre todo por ser tan buenas personas conmigo. Os quer´ıa dar las gracias tambi´en por visitarme en las estancias y preocuparos por m´ı. Ha sido un placer trabajar con vosotros. Amigrupodeinvestigaci´onporhacermesentircomounmiembrom´asdesde el primer d´ıa. Sois maravillosos. Tambi´en quer´ıa darle las gracias a los becarios, Henrique Barbosa, Tarcyana y Leandro por vuestro cari˜no y por todos los buenos momentos compartidos. Gracias a Cristian Morales y Francisco Javier Su´arez Grau por toda vuestra ayuda durante mi periodo de docencia. Finalmente, gracias a los miembros del Departamento de Ecuaciones Diferenciales y An´alisis Num´erico por el trato que me hab´eis dado durante estos cuatro a˜nos. A todas las maravillosas personas que conoc´ı durante mis dos estancias. A los profesores Martin Rasmussen y Michel Chipot por vuestra atenci´on y generosidad durante mis estancias. A Anna Cherubini y Cl´audia Buttarello Gentile, por todo el cari˜no y apoyo durante mi estancia en Londres. A Stephanie Zube, por tu generosidad y ayuda durante mi estancia en Zurich. A mis amigas, gracias por preocuparos por m´ı y apoyarme durante la elaboraci´on de este trabajo. A mi familia. Gracias a mis padres, Rosario y Manuel, por vuestro cari˜no, vuestra ayuda y comprensi´on en todo momento. Gracias por ense˜narme la importancia de trabajar y ser constante, valores por los que siempre os estar´e agradecida. Este trabajo tambi´en est´a dedicado a Raquel, Celia, Consuelo, Consolaci´on y Pablo, por vuestra paciencia y vuestros ´animos. Tambi´en quer´ıa dedicar este trabajo a Juan Manuel Ru´ız Labrador, que aunque tambi´en nos dej´o al principio de este camino, su constancia y fuerza de voluntad siempre han sido mi modelo a seguir. Finalmente, quer´ıa darle las gracias a Antonio Jes´us. Gracias por ser la mejor parte de m´ı, por tu cari˜no y apoyo incondicional. Por estar siempre en los malos momentos, nunca defraudarme y disfrutar mis alegr´ıas. Sinceramente gracias. 7 Contents Notation 11 Introduction 13 Spanish Summary 29 1 Abstract results on the theory of pullback attractors. Pullback flattening property 35 1.1 Basicconcepts............................... 36 1.2 Existence and relationships between pullback attractors . . . . . . . . 40 1.3 Pullback flattening property . . . . . . . . . . . . . . . . . . . . . . . 43 2 Non-autonomous nonlocal parabolic equation with sublinear terms 45 2.1 Statement of the problem. Existence results . . . . . . . . . . . . . . 46 2.2 Analysis of the stationary problem . . . . . . . . . . . . . . . . . . . 55 2.3 Minimal pullback attractors in L2-norm................. 59 2.4 Minimal attractors in H1-norm ..................... 67 3 Non-autonomous nonlocal reaction-di↵usion equations 75 3.1 Setting of the problem and existence results . . . . . . . . . . . . . . 76 3.2 Analysis of the stationary problem . . . . . . . . . . . . . . . . . . . 85 3.3 Existence of pullback attractors in L2(⌦)................ 94 3.4 Existence of pullback attractors in H1 0(⌦)................100 4 A monotone iterative approach for nonlocal reaction-di↵usion equations 109 4.1 Statement of the problem. Existence results . . . . . . . . . . . . . . 110 4.1.1 The monotonicity method for solving nonlinear PDEs . . . . . 111 4.1.2 Existence and uniqueness of weak solutions . . . . . . . . . . . 112 4.1.3 Strong solutions and regularising e↵ect.............116 4.2 Pullback attraction in L2-norm .....................119 4.3 Pullback attraction in H1-norm .....................128 5 Abstract results on the theory of multi-valued processes and pullback attractors 135 5.1 Basicconcepts...............................136 9 16 0 0.5 1 1.5 2 2.5 3 3.5 4 −0.5 0 0.5 1 1.5 2 2.5 µ2 y=l(\varphi)/s y=a(s) µ1 Figure 1: a(s)andl(') sintersect at two points. it holds (cf. [44, Theorem 4.1]) u(t)!u1strongly in L2(⌦) as t!1. Observe that when the function fappearing in problem (2) depends on the unknown u, the difficulty of the problem increases considerably, making the previous result not be available in this more complex framework. As far as we know, there is only a comparison result between the solution of the evolution problem and two (assumed to exist) stationary solutions (cf. Theorem 3.10). Later, Chipot & Molinet in [45] generalise the results obtained in [44], dealing with a continuum of steady states using dynamical systems. In the same lines of [44, 45], in [48] Chipot & Siegwart consider a more general elliptic operator than the and study the asymptotic behaviour of the solutions to problems with nonlocal di↵usion and mixed boundary conditions. In [35], Chang & Chipot are also interested in the asymptotic behaviour of the solutions to nonlocal problems, but in this case they deal with two nonlocal operators. In particular, they prove results which establish relationships between the solution to the evolution problem and stationary solutions. These results are similar to those given in the simpler framework of paper [44] which have been detailed previously. Considering also zero Dirichlet boundary conditions, in [50], Chipot & Zheng analyse the convergence of the solution of the evolution problem to one of the equilibria without assuming uniqueness of stationary solutions. Observe that not only have authors been interested in analysing problems in which the nonlocal operator is defined by a(l(u)), but they have also studied other variants, such as a(|ru|2)anda(krukp p). The first one @u @ta(|ru|2)u=f was analysed in [49] by Chipot et al. The main advantage of considering this new variation is that it allows to study the long-time behaviour of weak solutions making Introduction 17 use of global minimizers. In [47], Chipot & Savitska consider the p-Laplacian in the di↵usion term instead of the together with the nonlocal operator a(krukp p), which involve several difficulties, since although the p-Laplacian is a monotone operator, its lack of linearity makes it more complicated to deal with the nonlocal di↵usion as we will show later (see Chapter 7). Another variant, which has been analysed when fis still independent of u,deals with local nonlocal operators, i.e. the nonlocal operator is not acting globally in the whole domain but in a part of it contained in a ball centered on each position point: lr(·)(x): L2(⌦)!R u7! lr(u)(x)=ZBRN(x,r)\⌦ u(y)g(y)dy [g2L2(⌦)]. Problems with this kind of operators have been analised by Andami Ovono & Rougirel in [2, 3]. The authors study radial solutions, bifucation, the existence of branch of solutions and stability. Namely, in [2], Andami Ovono proves the existence of local branches of solutions in a radial setting by bifurcation analysis. Furthermore, in the cited paper, the existence of the compact global attractor in L2(⌦) is analysed. For fdepending on the unknown u, the situation is more involved. In [70], Hilhorst & Rodrigues analyse the parabolic equation @u @t=a✓1 |⌦|Z⌦ u(x)dx◆u+f✓u, 1 |⌦|Z⌦ u(x)dx◆, studying the existence of weak solutions and obtaining a rigorous derivation of a class of di↵usion equations that have been used to model the threshold phenomena in porous media combustion. Later, in [97], Menezes analyses the equation uta(l(u))u+f(u)=h(t)in⌦⇥(0,T), where fis a Lipschitz function, hbelongs to L2(0,T;H1(⌦)) and ⌦has a smooth boundary. Using fixed point techniques, the existence and uniqueness of weak solutions are analysed. Moreover, making use of the Galerkin approximations, the existence of periodic solutions is also studied. In this more complex framework of fdepending on u,Corrˆeaconsidersin[53] the nonlocal elliptic problem 8 < : a✓Z⌦ |u|q◆u=H(x)f(u)in⌦ u=0 on@⌦, and the existence of positive solutions through fixed point theorems is analysed. An analogous result is proved by Corrˆea et al. in [54], analysing the existence of positive solutions to 8 < : a✓Z⌦ u◆u=f(x, u)in⌦ u=0 on@⌦. 18 Recently, Figuereido de Sousa et al. have also made interesting contributions in this framework (see [58]), showing the existence of positive solutions to the non-local logistic equation 8 < : a✓Z⌦ u◆u=ub(x)u2in ⌦ u=0 on@⌦. As we can appreciate by the large number of references cited along this introduction, there have been many advances related to equation (1) and its variants, with contributions on existence and uniqueness of weak solutions, radial solutions, periodic solutions or convergence of the solution of the evolution problem towards a stationary solution, amongst others. Concerning the long-time behaviour of solutions, except for some concrete problems, it is an intractable task to study the existence of stationary solutions and their stability or Lyapunov functions, amongst others. That is why it is worth considering the information that can be obtained by the theory of attractors to study the asymptotic behaviour of the solutions (see [2], also some previous results on this direction by Lovat [87]). In the context of attractors, the compact global attractor is a useful tool that has been developed to study autonomous dynamical systems in the last few decades. Namely, this object has been deeply analysed by Hale [69], Ladyzhenskaya [83], Babin & Vishik [16], Vishik [113], Ball [17], Temam [111], Robinson [100], Sell & You [105] or Babin [15]. A global attractor is characterised for being a compact set in a given metric space which is maximal and invariant for the corresponding semiflow and attracts through the semiflow to all fixed nonempty bounded subsets of the metric space. In [69, 16, 100], the authors provided conditions that guarantee the existence of global attractors as well as examples. However, after including time-dependent terms, which allow to model more complex situations, studying the asymptotic behaviour of the solutions through the compact global attractor may not make much sense. For instance, consider the Cauchy problem (x0(t)=↵x(t)+t x(⌧)=x⌧,(4) where ↵>0.It is easy to check that the solution to (4) is defined by x(t;⌧,x⌧)=✓x⌧+1 ↵2⌧ ↵◆e↵(t⌧)+t ↵1 ↵2. Taking limit when t!1, we do not obtain any information about the behaviour of the solutions. What happens is that the trajectories are attracted by a time-dependent family defined by A={A(t)=t/↵1/↵2}.Forafixedt2R,thesectionA(t)isacompact set which attracts the solutions in a pullback sense, i.e. when the initial time ⌧!1.This approach allows us to establish not only the asymptotic behaviour of the dynamical system but also what the current attractions sections are when the Introduction 19 initial data come from long time ago in the past. Furthermore, observe that in the autonomous framework, both concepts, forward attraction and pullback attraction, coincide. In addition to this approach, there are other di↵erent ones from the point of view of non-autonomous dynamical systems, like skew-product flows (see Sell [104]) or uniform attractors and their kernel sections, which seem to be the natural generalisation of the compact global attractor, studying the asymptotic behaviour of the solutions when the time goes to infinity (see Chepizhov & Vishik [40]). All of them are valid to analyse di↵erent features of the evolution of a non-autonomous dynamical system. In this thesis, we choose the approach of pullback attractors (see Kloeden & Schmalfuß [81, 82] and Kloeden [76], also related to random dynamical systems [55]), since it allows us to minimize the assumptions on the forcing terms and the resultant objects are invariant (in a suitable “non-autonomous-dynamical-system sense”), unlike what happens with the uniform attractors which do not fulfil any property of invariance in general. However, this new object, the pullback attractor, may not be unique like the compact global attractor. For instance, see [93, Example 11]. In the cited example, Mar´ın-Rubio & Real consider a continuous function f= f(t, x):R27! R, which is globally Lipschitz w.r.t. xand fulfils that f(t, x)=xif |x|et.Then,theydefineaprocessUon Rgiven by U(t, ⌧)x⌧=x(t;⌧,x⌧)8x⌧2R8t⌧, where x(t;⌧,x⌧) is the unique solution to the Cauchy problem (x0(t)=f(t, x) x(⌧)=x⌧. To prove that there exist more than one pullback attractor consider A1={0}and A2={A2(t)=[et,e t]:t2R}. Both of them are families of compact and nonempty sets, invariant for the process Uin a non-autonomous-dynamical-system sense, and attract fixed nonempty bounded subsets of R(for more detail about pullback attractors see Chapter 1). To solve this problem, a minimality condition is imposed (cf. Definition 1.11). This way the uniqueness is guaranteed, being A1 the minimal pullback attractor. In this approach of pullback attractors, many new results have appeared over the last years. Some authors have been interested in studying the pullback attractor in the classical sense, i.e. the pullback attractor of solutions starting in fixed bounded sets. Others, though, have employed the concept of attraction related to a class of families, called universe D,whichismadeupofsetswhichareallowedtomovein time and are usually defined in terms of a tempered condition (e.g. cf. [51, 31, 32]). In [93], Mar´ın-Rubio & Real analyse these two di↵erent concepts of pullback attractor with detail, highlighting some difficulties which appear in the framework of the universe of fixed nonempty bounded sets and which can be solved making use 20 of tempered tools. Finally, the authors establish relationships between these two notions of attractors proving that in fact both families coincide under a suitable assumption (see [93, Proposition 23]). This PhD project is split into seven chapters. In Chapter 1, abstract results on pullback attractors within the framework of universes are analysed. Chapters 2, 3 and 4 are devoted to studying the existence and uniqueness of solutions as well as the existence of minimal pullback attractors in the phase spaces L2(⌦) and H1 0(⌦) for non-autonomous nonlocal parabolic equations. Next, in Chapter 5, the theory of pullback attractors for multi-valued non-autonomous dynamical systems will be analysed to be applied later in the following two chapters, Chapters 6 and 7. In Chapter 6, a non-autonomous nonlocal reaction-di↵usion equation with a small perturbation in the nonlocal di↵usion term and the non-autonomous force is analysed in a multi-valued framework. The existence of weak solutions and pullback attractors in L2(⌦) is proved. In addition, the upper semicontinuous behaviour of attractors will be analysed when the perturbation goes to zero. Finally, in Chapter 7, the existence of solutions for an autonomous nonlocal p-Laplacian equation is shown. Furthermore, the asymptotic behaviour of the solutions is studied proving the existence of the compact global attractor in L2(⌦).The study of problem ( e P) in the non-autonomous framework is also possible, but for the sake of simplicity, we have focused on a problem without non-autonomous terms to make the proof clearer. Chapter 1 is split in three sections. In Section 1.1, we consider a universe D composed of families parameterised in time. We analyse some basic concepts as well as some abstract results which will be crucial to prove the existence of pullback attractors under minimal assumptions. In Section 1.2, we study the main result of this chapter. Namely, Theorem 1.13 guarantees the existence of the minimal pullback D-attractor. Making use of this result, in Corollary 1.15, we establish relationships between the attractor of the universe of fixed nonempty bounded sets and the attractor associated to the universe D(for more details see [93, Proposition 23]). In addition, in Theorem 1.16, we also establish relationships between attractors related to general universes associated to di↵erent phase spaces (see [62, Theorem 3.15]). To end this chapter, in Section 1.3, the flattening property is analysed. The notion provided in this thesis is a slight modification of the well-known “Condition (C)” introduced by Ma, Wang and Zhong in [88], coined by Kloeden & Langa as the flattening property in [77]. It is a useful tool that allows to prove one of the main ingredients in order to guarantee the existence of the minimal pullback attractor, the pullback asymptotic compactness (cf. Proposition 1.18). In Chapter 2, the non-autonomous nonlocal parabolic problem (P)8 > < > : @u @ta(l(u))u=f(u)+h(t)in⌦⇥(⌧,1), u=0 on@⌦⇥(⌧,1), u(x, ⌧)=u⌧(x)in⌦, Introduction 21 is analysed, where ⌦is an open bounded subset of RN,⌧2Rand the function a2C(R;R+) is locally Lipschitz and there exists a positive constant m, such that 0<ma(s)8s2R.(5) In addition, l2L(L2(⌦),R), f2C(R)andthereexisttwoconstants⌘>0and Cf0 such that |f(s)|Cf(1 + |s|)8s2R, (f(s)f(r))(sr)⌘(sr)28s, r 2R.(6) To conclude the setting of (P), we assume that the initial datum u⌧2L2(⌦) and the non-autonomous term h2L2 loc(R;H1(⌦)). We divide Chapter 2 into four sections. Section 2.1 is devoted to existence and uniqueness results. First of all, the existence of local solution to (P) is proved making use of [52, Theorem 1.1, p.43], which is a generalisation of the Peano Theorem. To guarantee the uniqueness, the function aalso needs to be locally Lipschitz. Then, in Theorem 2.4, the existence and uniqueness of weak (global) solutions are analysed making use of the Galerkin approximations and compactness arguments. The main difficulty of this result consists in dealing with the limit of the Galerkin approximations of the nonlinear terms a(l(u))uand f(u). To do this, in addition to applying the Aubin-Lions lemma, we use [85, Lemme 1.3, p. 12], which allows us to work with the nonlinearities. Furthermore, in this section, namely in Theorem 2.5, the regularising e↵ect of the equation is stated as well as the existence of strong solution in a more regular framework. In Section 2.2, we study the existence and uniqueness of stationary solutions and their stability. For the existence result, we make use of a corollary of the Brouwer fixed point theorem (see [85, Lemme 4.3, p. 53]). Furthermore, the uniqueness of stationary solution is shown under additional requirements. The exponential decay of the solution of the evolution problem (P) towards the unique stationary solution is also analysed. Section 2.3 is devoted to proving the existence of minimal pullback attractors in the phase space L2(⌦) as well as some relationships between these families (cf. Theorem 1.15). In this case, to build a suitable tempered universe for our purposes, we need to make this additional assumption on the function f lim sup |s|!1 f(s) s<m 1,(7) where 1is the first eigenvalue of with zero Dirichlet boundary conditions. Observe that to prove the most difficult result of this section, the pullback asymptotic compactness (cf. Proposition 2.15), which leads immediately to the existence of attractors, we use an energy method which relies on the continuity of the solutions (cf. for more details [73, 92, 94, 62]). Finally, in Section 2.4, in a more regular framework we show the existence of minimal pullback attractors in the phase space H1 0(⌦) and establish some relationships between these objects and the attractors analysed in Section 2.3. In this case, 22 the pullback asymptotic compactness is also proved using the energy method that has been applied in Section 2.3, but making use of a more regular energy equality associated to strong solutions (cf. Proposition 2.22). All the results of this chapter can be found in [21]. Chapter 3 is devoted to studying problem (P) when the function f2C(R) satisfies (6) and there exist positive constants ↵1,↵2,and p>2 such that ↵1|s|pf(s)s↵2|s|p8s2R.(8) Although we relax the assumptions on f, since now we are not dealing with semilinear reaction terms, we need to impose some strong smoothness condition on the domain ⌦. Nevertheless, we do not impose any restriction on the dimension of the domain ⌦⇢RN, which allows to deal with problems which have a strong dependence not only on the spatial variables, but also on others. The structure of Chapter 3 is as follows. In Section 3.1, the existence and uniqueness of weak solutions to (P) is analysed in Theorem 3.3. To do this, we use the Galerkin approximations and compactness arguments, together with some projection operators which are well-defined thanks to the regularity imposed to the domain ⌦. Next, the existence and uniqueness of strong solutions to (P) and the regularising e↵ect of the equation are studied in Theorem 3.4. Whereas in local reaction-di↵usion equations, strong solutions belong to L2(⌧,T;H2(⌦)\H1 0(⌦))\C([⌧,T]; H1 0(⌦)) for all T>⌧(cf. [100, 5]), in nonlocal problems like problem (P) we are not able to obtain the regularity C([⌧,T]; H1 0(⌦)) for the solution u, due to the fact that in general, u02Lq(⌧,T;Lq(⌦)) (1 q<2), instead of belonging to L2(⌧,T;L2(⌦)). Therefore, making use of the fact that u2L1(⌧,T;H1 0(⌦)) together with the cited regularity of u0, we can only obtain that u2Cw([⌧,T]; H1 0(⌦)) for all T⌧(cf. [108, Theorem 2.1, p. 544] or [111, Lemma 3.3, p. 74]). Furthermore, a Maximum Principle is provided for problem (P), which is an essential tool to study biological models, because in order to analyse population dynamics it is crucial to guarantee that the solution is positive (for more details see Remark 3.25). Section 3.2 is in a certain sense complementary to our main results on attractors. We analyse some results concerning the stationary solutions. Observe that due to the presence of the nonlinear terms a(l(u))uand f(u), this problem is far from being trivial. Therefore we can only provide some partial results. Namely, we establish the existence of nontrivial solutions for a special choice of fmaking use of a result by Chipot & Corrˆea (cf. [42, Theorem 2.1]). In a more general framework, we provide a conditional result in the same line as [44, Lemma 4.1]. Namely, we obtain a comparison result between the solution to problem (P) and two (assumed to exist) stationary solutions. Section 3.3 focuses its study on the existence of minimal pullback attractors in the phase space L2(⌦) and some relations between them are obtained. Regarding f, while in the sublinear framework (see Chapter 2) we also needed to impose the usual assumption (7) to study the asymptotic behaviour of the solutions through the theory of attractors, here condition (8) will be enough. Introduction 23 To conclude this chapter, Section 3.4 is devoted to analysing the existence of minimal pullback attractors in the phase space H1 0(⌦) in the strong-solutions framework. Due to the fact that in general f(u)belongstoLq(⌧,T;Lq(⌦)), it does not make any sense to multiply the equation of problem (P) by u2L2(⌧,T;L2(⌦)). Therefore, in the general case in which f(u)doesnotbelongtoL2(⌧,T;L2(⌦)), we cannot use an energy equality in a strong sense. However, to prove the existence of pullback attractors in H1 0(⌦), we need to deal with this kind of equalities. That is the reason why in this last section we impose the additional assumption f(u)2L2(⌧,T;L2(⌦)) 8u2L2(⌧,T;H2(⌦)\H1 0(⌦)) \L1(⌧,T;H1 0(⌦)), which will be replaced along this section by kf(u)k2 L2(⌧,T;L2(⌦)) Cfkuk2˜ b L1(⌧,T;H1 0(⌦))kuk2ˆ b L2(⌧,T;H2(⌦)\H1 0(⌦)), for some ˆ b,˜ b,Cf>0. This assumption has been obtained using interpolation results (cf. [116, Lemma II.4.1, p. 72]) and the regularity of the domain ⌦. Then, taking this into account, the existence of attractors is guaranteed, where the pullback asymptotic compactness has been proved using the same kind of energy method as the one applied in Chapter 2 for the same purpose. All the results of these chapter have been analysed in [23]. In Chapter 4, we continue analysing problem (P) in the reaction-di↵usion framework, i.e. when the function ffulfils the assumptions (6) and (8), like in Chapter 3. In this chapter 4, we get rid of the strong assumptions made on the domain ⌦in Chapter 3, which allows to model real phenomena with more accuracy since they tend to be posed in nonsmooth domains (see [68] for more details). Nevertheless, in this case we need to impose some restrictions on either the dimension N (cf. Theorem 4.8) or the reaction term (see Theorem 4.10) or even both of them (cf. Corollary 4.11) when we are analysing the existence of strong solutions and the regularising e↵ect of the equation, unlike what happened in Chapter 3 that it was only necessary smoothness assumptions on the domain ⌦. To prove the existence of pullback attractors in H1 0(⌦) in this chapter 4, it is not enough with assuming only the restrictions on the dimension of the domain ⌦that were made in Theorem 4.8, we need to study the existence of these families in the settings of Theorem 4.10, imposing requirements on the reaction term, or Corollary 4.11, making assumptions on the dimension of the domain and the reaction term. Observe that thanks to the restrictions made on the dimension of the domain in Corollary 4.11, the reaction term can be taken more general in this corollary than in Theorem 4.10. We split this chapter into three sections. In Section 4.1, we study the existence of weak and strong solutions to the nonlocal reaction-di↵usion problem (P). In the first part, Section 4.1.1, we briefly recall the monotonicity method for solving nonlinear PDEs (cf. [85, Chapitre 2]). Next, in Section 4.1.2, the existence and uniqueness of weak solutions to (P) are shown making use of an iterative method together with monotonicity and compactness arguments. Namely, we consider the sequence of 24 problems (Pn)8 > > > > < > > > > : u2L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) \C([⌧,T]; L2(⌦)) 8T⌧, d dt(u(t),v)+a(l(un1(t)))((u(t),v)) = (f(u(t)),v)+hh(t),vi, u(⌧)=u⌧, where u0⌘0andunis the solution to (Pn)ifn1. Observe that the existence and uniqueness of weak solutions to (Pn) are guaranteed by the monotonicity method. Then, applying compactness arguments to the sequence {un},wecanprovetheexistenceofweaksolutionsto(P).Observethat this result is an improvement compared to the existence result of Chapter 3 in the weak-solutions framework (cf. Theorem 3.3), since in this case we are able to prove the existence of weak solutions to (P) without making any smoothness assumptions on the domain ⌦. The uniqueness holds immediately when we also assume that the function ais locally Lipschitz. Next, in Section 4.1.3, the existence and uniqueness of strong solutions as well as the regularising e↵ect of the equation are proved without assuming any smoothness conditions on the domain ⌦as in Chapter 3. In return, we need to impose some restrictions on either the dimension N(cf. 4.11) like in Theorem 4.8 or the reaction term. Namely in Theorem 4.10, to prove the existence and uniqueness of strong solutions and the regularising e↵ect of the equation, we assume that |f(s)|C(1 + |s|+1)8s2R,(9) with =2/N if N3, where this estimate has been obtained applying interpolation results (cf. [116, Lemma II.4.1, p. 72]) to the Sobolev spaces L1(⌧,T;L2(⌦)) and L2(⌧,T;H1 0(⌦)). Observe that when N=1,2, can be any positive value (cf. Remark 4.9 (i)). In fact, the assumption (9) can be improved if we impose some requirements on the dimension of the domain ⌦. Namely, =2/(N2) when 3N2p/(p2) (see Corollary 4.11 for more details). Section 4.2 is devoted to studying the existence of pullback attractors in the phase space L2(⌦) in the more general setting of Section 4.1.2. Although this result is not new in this PhD project, since the existence of these families has been proved in Chapter 3 (see Theorem 3.17 for more details), the method applied to prove the pullback asymptotic compactness is. Namely, we argue similarly as it was done in [101]. For the sake of completeness, in Section 4.3, we analyse the existence of pullback attractors in H1 0(⌦) in the framework of universes and establish some relationships amongst these families of attractors and the ones analysed in Section 4.2. Observe that the existence of pullback attractor in H1-norm has been analysed in Chapter 3 (cf. Theorem 3.23). Nevertheless, in this chapter to prove the pullback asymptotic compactness we use the pullback flattening property, which is a tool that has not been used before in this thesis. In addition, it is worth highlighting that to prove these results, we need to work in the setting of Theorem 4.10 or Corollary 4.11, since assuming only restrictions on the dimension of the domain ⌦is not enough to guarantee the existence of stronger energy equalities as well as the continuity of the Introduction 25 strong solutions in H1 0(⌦).The results of this chapter can be found in [25]. From Chapter 5 forward, all the results provided in this PhD project are set in a multi-valued framework. Many authors have been interested in analysing problems in which the uniqueness of solution is not guaranteed. Amongst them, it is worth highlighting 3D incompressible Navier-Stokes equations (cf. [17]), di↵erential inclusions (cf. [96]), reaction-di↵usion equations (cf. [6]) and delay di↵erential equations (cf.[89]). In Chapter 5, we describe abstract results on multi-valued non-autonomous dynamical systems in the framework of universes. This chapter is divided into two sections. In Section 5.1, we analyse the basic concepts studied in Chapter 1 in this new setting. In addition, the notion of upper semicontinuous process (cf. Definition 5.2) is shown. Later, in Section 5.2, the existence of pullback attractors and some relationships between them are established in this multi-valued framework (cf. Theorem 5.11, Corollary 5.13 and Theorem 5.14). Chapter 6 is devoted to the study of the perturbed non-autonomous nonlocal reaction-di↵usion problem (P")8 > < > : @u @t(1 ")a(l(u))u=f(u)+"h(t)in⌦⇥(⌧,1), u=0 on@⌦⇥(⌧,1), u(x, ⌧)=u⌧(x)in⌦, where "2[0,1), ⌧2R,⌦⇢RNis a bounded open set, the function a2C(R;R+) fulfils (5), f2C(R) satisfies (8), and l2L(L2(⌦),R).In addition, the initial datum u⌧2L2(⌦) and the non-autonomous term h2L2 loc(R;H1(⌦)). This chapter is split into three sections. In Section 6.1, the existence of weak solutions to (P") is analysed in Theorem 6.2 making use of the Galerkin approximations and compactness arguments. Observe that although we are analysing a reaction-di↵usion equation similar to the one studied in (P), it is not necessary to assume any smoothness condition on the domain ⌦. The reason is that along the proof of this result, we do not need to obtain a uniform estimate of the Galerkin approximations associated to u0, the temporal derivative of a solution uto (P"), since instead of applying the Aubin-Lions lemma, we make use of the compactness by translation (cf. [110, Theorem 13.2, p. 97] and [110, Remark 13.1, p. 100]) in the compactness arguments. In Section 6.2, namely in Theorem 6.12, the existence of minimal pullback attractors in the phase space L2(⌦) is stated and some relationships between them are established. In this case, to prove that the multi-valued process Uis asymptotically compact, we make use of the same kind of energy method applied in Chapters 2 and 3 for the same purpose. In Section 6.3, we study the upper semicontinuous behaviour of attractors. In Theorem 6.15, we prove that the family of pullback attractors indexed by "converges to the global attractor associated to (P0) when the parameter "goes to zero. 32 dominio (v´ease el Teorema 3.3). Adem´as analizamos la existencia y unicidad de soluci´on fuerte as´ı como el efecto regularizante de la ecuaci´on. Obs´ervese que mientras que en las ecuaciones de reacci´on-difusi´on locales, la soluci´on fuerte upertenece a L2(⌧,T;H2(⌦)\H1 0(⌦)) \C([⌧,T]; H1 0(⌦)) para todo T>⌧(cf. [100, 5]), en las variantes no locales analizadas en este cap´ıtulo no podemos alcanzar en general la regularidad C([⌧,T]; H1 0(⌦)),debido a que u02Lq(⌧,T;Lq(⌦)) (donde 1 q<2), en lugar de pertenecer a L2(⌧,T;L2(⌦)), que junto con u2L2(⌧,T;H2(⌦)\H1 0(⌦)), garantizar´ıan la citada continuidad en H1 0(⌦) de la soluci´on. A continuaci´on, en la Secci´on 3.2, realizamos algunas aportaciones en el marco estacionario. Concretamente, en los Teoremas 3.8 and 3.10 estudiamos respectivamente la existencia de soluciones no triviales y un resultado de comparaci´on entre la soluci´on del problema evolutivo y dos soluciones estacionarias. Finalmente, en las dos ´ultimas secciones analizamos el comportamiento asint´otico de las soluciones a trav´es de la teor´ıa de atractores pullback. En la Secci´on 3.3 demostramos la existencia de atractores pullback en L2(⌦) en el marco de los universos y establecemos relaciones entre estas familias (v´ease el Teorema 3.17). Para concluir este cap´ıtulo, en la Secci´on 3.4, estudiamos la existencia de atractores pullback en H1 0(⌦). Para ello, necesitamos imponer hip´otesis adicionales sobre la funci´on fque garanticen que f(u)perteneceaL2(⌧,T;L2(⌦)). Estas hip´otesis son construidas usando resultados de interpolaci´on y la regularidad del dominio ⌦. De esta forma, u02L2(⌧,T;L2(⌦)) y por tanto, las manipulaciones con utienen sentido as´ı como es posible obtener estimaciones m´as regulares usando la igualdad de energ´ıa asociada a las soluciones fuertes (cf. (3.55)). Estos resultados han sido analizados en el trabajo [23]. En el Cap´ıtulo 4 continuamos analizando la ecuaci´on de reacci´on-difusi´on no local con t´erminos no aut´onomos estudiada en el Cap´ıtulo 3 bajo otras condiciones. A lo largo de este cap´ıtulo no imponemos ninguna regularidad al dominio ⌦. Esto nos permite modelar problemas reales con m´as precisi´on ya que muchos de ellos est´an planteados en dominios no regulares (v´ease [68] para m´as detalles). Este cap´ıtulo est´a dividido en tres secciones. En la Secci´on 4.1 analizamos la existencia de soluciones d´ebiles y fuertes. A diferencia de lo que ocurr´ıa en el Cap´ıtulo 3, en el Teorema 4.6 se demuestra la existencia y unicidad de soluci´on d´ebil sin imponer ninguna regularidad al dominio, utilizando un m´etodo iterativo y argumentos de compacidad y de monoton´ıa. Este resultado supone una mejor´ıa con respecto al resultado de existencia de soluci´on d´ebil del Cap´ıtulo 3 (cf. Teorema 3.3), en el que la regularidad del dominio ⌦era imprescindible. La existencia y unicidad de soluci´on fuerte as´ı como el efecto regularizante de la ecuaci´on se prueba haciendo uso de las aproximaciones de Galerkin y argumentos de compacidad. Como no asumimos que el dominio ⌦es regular, como se hizo en el Cap´ıtulo 3, a cambio necesitamos imponer ciertas restricciones a la dimensi´on del dominio N(cf. Teorema 4.8), al t´ermino de reacci´on (cf. Teorema 4.10) o a ambos (cf. Corolario 4.11). A continuaci´on, en la Secci´on 4.2, estudiamos la existencia de atractores pullback en L2(⌦) en el marco de los universos. Aunque este resultado no es nuevo en esta Spanish Summary 33 tesis, la existencia de estas familias ha sido probada en el Cap´ıtulo 3 (cf. Teorema 3.17), el m´etodo usado para probar la compacidad asint´otitca pullback s´ı lo es. Concretamente, se emplea el m´etodo usado por Rosa en [101] adaptado al marco no local. Finalmente, en la Secci´on 4.3, analizaremos la existencia de atractores pullback en H1 0(⌦). Mientras que en el Cap´ıtulo 3, para garantizar que f(u)2L2(⌧,T;L2(⌦)) en el marco de las soluciones fuertes us´abamos que el dominio ⌦fuese regular, en este cap´ıtulo para obtener que f(u)2L2(⌧,T;L2(⌦)), como no estamos asumiendo ninguna condici´on de regularidad al dominio ⌦, tenemos que imponer algunas restricciones al t´ermino de reacci´on (cf. Teorema 4.10). Obs´ervese que estos requerimientos pueden ser debilitados imponiendo ciertas restricciones a la dimensi´on del dominio ⌦(cf. Corolario 4.11). Los problemas analizados en los restantes cap´ıtulos de esta tesis, concretamente en los Cap´ıtulos 6 and 7, est´an planteados en un marco multivaluado, ya que no podemos garantizar la unicidad de soluci´on bajo las hip´otesis impuestas. Por ello, en el Cap´ıtulo 5, estudiamos algunos resultados abstractos sobre sistemas din´amicos multivaluados para procesos. Este Cap´ıtulo 5 est´a constituido por dos secciones. En la Secci´on 5.1 definimos algunos conceptos b´asicos y estudiamos varios resultados abstractos que nos permitir´an demostrar el teorema principal de existencia de atractores pullback en la Secci´on 5.2. Concretamente, dicho resultado se corresponde con el Teorema 5.11. A continuaci´on, en el Corolario 5.13 se establecen relaciones entre el atractor de los acotados fijos y el atractor asociado a un universo Dconstituido por familias parametrizadas en tiempo. Para concluir el cap´ıtulo, en el Teorema 5.14, estudiamos m´as relaciones que se pueden establecer entre atractores asociados a universos m´as generales. En el Cap´ıtulo 6, estudiamos una ecuaci´on de reacci´on-difusi´on no local sin unicidad de soluci´on con una peque˜na perturbaci´on "en el t´ermino de difusi´on y en la fuerza no aut´onoma. Este cap´ıtulo est´a dividido en tres secciones. En la Secci´on 6.1 demostramos la existencia de soluciones d´ebiles usando las aproximaciones de Galerkin y argumentos de compacidad. Obs´ervese que a diferencia de lo que ocurr´ıa en el Cap´ıtulo 3, en este caso no es necesario imponer ninguna regularidad al dominio ⌦. Esto es debido a que a lo largo de la prueba del Teorema 6.2 no realizamos ninguna estimaci´on uniforme de la aproximaci´on de Galerkin asociada a la derivada temporal de una soluci´on, ya que en los argumentos de compacidad, en lugar de usar el lemma de Aubin-Lions, hacemos uso de la compacidad por traslaci´on (cf. [110, Theorem 13.2, p. 97] y [110, Remark 13.1, p. 100]). A continuaci´on, en la Secci´on 6.2, demostramos la existencia de atractores pullback en L2(⌦). Para ello, en la Proposici´on 6.11 se analiza la compacidad asint´otica pullback aplicando el mismo tipo de m´etodo de energ´ıa que el usado en los Cap´ıtulos 2y3. En la Secci´on 6.3, estudiamos la propiedad de semicontinuidad superior de atractores. Concretamente, en el Teorema 6.15 se prueba que la familia de atractores pullback dependiente del par´ametro ",cuyaexistenciahasidodemostradaenel 34 Teorema 6.12, converge, cuando el par´ametro tiende a cero, al atractor global del semiflujo multivaluado asociado al problema aut´onomo inicial con "=0. Para finalizar este cap´ıtulo, en la Secci´on 6.4, estudiamos algunos resultados de regularidad. Concretamente, estudiamos la existencia de soluciones fuerte para (P")as´ıcomoelefectoregularizantedelaecuaci´on.Acontinuaci´on,demostramos la existencia de atractores pullback en H1 0(⌦) as´ı como generalizamos el resultado de semicontinuidad superior de atractores estudiado en la secci´on anterior, demostrando la convergencia en la H1-norma. Para estudiar el comportamiento de las soluciones en este marco m´as regular, como la unicidad de soluci´on no est´a garantizada, usamos un razonamiento de regularidad a posteriori (v´ease el Teorema 6.17). Los resultados de este cap´ıtulo han sido analizados en los trabajos [22, 26]. En el Cap´ıtulo 7, analizamos un problema aut´onomo en el que el t´ermino de difusi´on est´a constituido por un operador no local y el p-Laplaciano, generalizando as´ı la difusi´on con respecto a los cap´ıtulos anteriores, en los que todos los an´alisis han sido hechos para el Laplaciano. Este cap´ıtulo est´a dividido en dos secciones. En la Secci´on 7.1 probamos la existencia de soluciones d´ebiles. A trav´es de un cambio de variable temporal transformamos un problema con difusi´on no local en un problema con difusi´on local, an´alogamente a como fue hecho por Chipot & Savitska en [47]. Sin embargo, la unicidad de soluci´on no est´a garantizada debido a la generalidad del operador no local. Adem´as de la existencia de soluciones d´ebiles, en el Teorema 7.2, se demuestra una propiedad de regularizaci´on del problema analizado. A continuaci´on, en la Secci´on 7.2, analizamos el comportamiento asint´otico de las soluciones demostrando la existencia del atractor global en L2(⌦) ya que el problema estudiado no posee t´erminos no aut´onomos. Para ello, en la Proposici´on 7.8 construimos un conjunto absorbente en W1,p 0(⌦) empleando la propiedad regularizante citada anteriormente. Finalmente, teniendo esto en cuenta junto con la compacidad de la inyecci´on W1,p 0(⌦),!L2(⌦),concluimos este trabajo demostrando la existencia del atractor global en el Teorema 7.9. Obs´ervese que tambi´en es posible analizar el problema ( e P) en el marco no aut´onomo. Sin embargo, por simplicidad el estudio se ha hecho sin t´erminos dependientes del tiempo, ya que la prueba del Teorema 7.2 emplea un cambio de variable que la hace muy t´ecnica. Estos resultados han sido analizados en [24]. Para concluir este proyecto de tesis, proporcionamos una lista no exhaustiva de problemas que nos gustar´ıa estudiar en el futuro como por ejemplo, ecuaciones no locales en dominios no acotados, la propiedad Kneser o problemas no locales con retardo, entre otros. Adem´as, describimos algunos de los trabajos en curso y resaltamos las dificultades que nos han aparecido. Chapter 1 Abstract results on the theory of pullback attractors. Pullback flattening property The modelisation of real phenomena in di↵erent scientific fields like Physics, Biology or Chemistry, makes the equations more and more complex when they try to reproduce the reality with accuracy. As a consequence, the study of the existence of points of equilibrium and their stability or Lyapunov functions associated to partial di↵erential equations proves to be an intractable task in many occasions. On the other hand, as a natural generalisation of the behaviour of the solutions around points of equilibrium and thanks to the presence of chaos and turbulence phenomena in the reality, the dynamical systems field, which involves the theory of attractors, inertial manifolds or fractal dimension analysis in diverse senses, amongst others, has been developing in the last few decades. In the context of attractors there are several choices to study the asymptotic behaviour of the solutions of evolution problems. One can prove the existence of the global compact attractor in the autonomous framework (cf. [100]). However, when the equation possesses time-dependent terms, several approaches from nonautonomous dynamical systems can be used. Namely, one can do attempts with uniform attractors (cf. [40]), skew-product flows (cf. [105]) and pullback attractors (see [76, 31, 32, 93, 62] for more details; also related to random dynamical systems, cf. [55]). In this chapter, we analyse abstract results of the theory of pullback attractors, which allow us to study not only the future of the dynamical system but also what the current attracting sections are when the initial data come from 1. In addition, making use of this approach, we can analyse the existence of attractors for equations with general forcing terms and the resultant objects are invariant in a “suitable non-autonomous-dynamical-system sense”, unlike what happens with the uniform attractors. The theory of pullback attractors has been used for a wide range of problems such as non-autonomous di↵erence equations (cf. [76]), non-autonomous and stochastic multi-valued dynamical systems (cf. [29]), non-autonomous di↵erence inclusions (cf. [79]), non-autonomous 2D-Navier-Stokes equations in some unboun- 35 36 1.1. Basic concepts ded domains (cf. [32, 84]), non-autonomous di↵erential equations (cf. [14]), nonautonomous reaction-di↵usion equations in unbounded domains (cf. [114, 7, 6, 115]), 2D or 3D-Navier-Stokes equations with delay (cf. [95, 90, 63, 64, 66]). Within this framework, some authors are interested in studying the pullback attractor in the classical sense, i.e. the pullback attractor of solutions starting in “fixed” bounded sets. Others, though, employ the concept of attraction related to a class of families, called universe D,madeupbysetswhichareallowedtomoveintimeandusually defined in terms of a tempered condition (cf. [31, 32, 61]). This approach will be the one analysed along this chapter. The results of this chapter can be found in [93, 62, 33, 65]. 1.1 Basic concepts Consider given a metric space (X, dX). Definition 1.1. (a) A process on X(also called a two-parameter semigroup) is a mapping R2 d⇥X3 (t, ⌧, x)7! U(t, ⌧)x2Xsuch that U(⌧,⌧)x=xfor any (⌧,x)2R⇥X, and U(t, s)(U(s, r)x)=U(t, r)xfor any rstand all x2X. (b) A process Uon Xis said to be 1. continuous if for any pair (t, ⌧)2R2 d, the mapping U(t, ⌧):X!Xis continuous; 2. strong-weak (also known as norm-to-weak) continuous if for any pair (t, ⌧)2R2 d,the map U(t, ⌧)is continuous from Xwith the strong topology into Xwith the weak topology; 3. closed if for any pair (t, ⌧)2R2 dand any sequence {xn}⇢X, if xn! x2Xand U(t, ⌧)xn!y2X, then U(t, ⌧)x=y. Remark 1.2. It is clear that every continuous process is strong-weak continuous and every strong-weak continuous process is closed. Let b D0={D0(t):t2R}⇢P(X)beafamilyofnonemptysets. Observethat we do not require any additional condition on these sets such as compactness or boundedness. Definition 1.3. A process Uon Xis said to be pullback b D0-asymptotically compact if for any t2R, and any sequences {⌧n}⇢(1,t]and {xn}⇢Xsatisfying ⌧n!1and xn2D0(⌧n)for all n, the sequence {U(t, ⌧n)xn}is relatively compact in X. Denote the omega-limit set of b D0by ⇤(b D0,t)=\ st[ ⌧s U(t, ⌧)D0(⌧) X 8t2R.(1.1) Chapter 1. Theory of Pullback Attractors 37 Proposition 1.4 (Sequential characterisation of the omega-limit set).It holds that y2⇤(b D0,t)if and only if there exist sequences {⌧n}⇢(1,t]and {xn}⇢X, with ⌧n!1and xn2D(⌧n)for all n, such that U(t, ⌧n)xn!y. Then, we have the following result. Proposition 1.5. If the process Uon Xis pullback b D0-asymptotically compact, then for any t2R, the set ⇤(b D0,t)given by (1.1) is a nonempty compact subset of Xand lim ⌧!1 distX(U(t, ⌧)D0(⌧),⇤(b D0,t)) = 0.(1.2) In addition, the family {⇤(b D0,t):t2R}is minimal in the sense that if b C={C(t): t2R}⇢P(X)is a family of closed sets such that lim ⌧!1 distX(U(t, ⌧)D0(⌧),C(t)) = 0, then ⇤(b D0,t)⇢C(t). Proof. Consider fixed t2R,andsequences{⌧n}⇢(1,t]and{xn}⇢Xsuch that ⌧n!1and xn2D0(⌧n)foralln.SincetheprocessUis pullback b D0- asymptotically compact, there exist subsequences {⌧n}and {xn}(relabeled the same) and y2Xsuch that U(t, ⌧n)xn!yin X.Then,fromthesequentialcharacterization of ⇤(b D0,t), it holds that y2⇤(b D0,t). Therefore, ⇤(b D0,t)isnonempty. Now we will show that the set ⇤(b D0,t)iscompact. Todothis,since⇤( b D0,t) is closed (cf. (1.1)), we only need to prove that this set is relatively compact in X.Tothatend,consider{yn}⇢⇤(b D0,t). Since yn2⇤(b D0,t)foralln,fromthe sequential characterization of ⇤(b D0,t), we deduce that there exist ⌧ntnand xn2D0(⌧n)suchthat dX(yn,U(t, ⌧n)xn)1 n.(1.3) Now, taking into account that the process Uis pullback b D0-asymptotically compact, it holds that there exists a convergent subsequence of {U(t, ⌧n)xn}. Then, from (1.3), there exists a convergent subsequence of {yn}.Thus,⇤( b D0,t)isacompactset. Thereupon, to prove (1.2), we argue by contradiction. Assume that there exists t2Rsuch that (1.2) does not hold. Then, there exist ">0 and sequences {⌧n}⇢ (1,t]and{xn}⇢Xwith ⌧n!1and xn2D0(⌧n), such that dX(U(t, ⌧n)xn,⇤(b D0,t)) >"8n1.(1.4) On the other hand, since the process Uis pullback b D0-asymptotically compact, there exists a subsequence of {U(t, ⌧n)xn}which converges to an element of ⇤(b D0,t), which is a contradiction with (1.4). Finally, we will prove that given a family of closed sets b C={C(t):t2R}⇢ P(X) that fulfils lim ⌧!1 distX(U(t, ⌧)D0(⌧),C(t)) = 0,(1.5) 38 1.1. Basic concepts the relationship ⇤(b D0,t)⇢C(t) holds. Consider fixed t2R,">0andx2⇤(b D0,t). Let us see that x2C(t).To do this, we will prove that BX(x, ")\C(t)6=;.(1.6) Since x2⇤(b D0,t), from the sequential characterization of ⇤(b D0,t), it holds that there exist sequences {⌧n}⇢(1,t]and{xn}⇢Xwith ⌧n!1and xn2D0(⌧n) for all n,suchthat lim n!1 dX(x, U(t, ⌧n)xn)=0. Therefore, there exists n0(")1 such that dX(x, U(t, ⌧n)xn)<" 28nn0(").(1.7) On the other hand, from (1.5) we deduce that there exists n1(")1 such that dX(U(t, ⌧n)xn,C(t)) <" 28nn1(").(1.8) Then, taking into account (1.7) and (1.8), (1.6) holds. Now, if we assume that the process Uis closed, the invariance of the family of sets {⇤(b D0,t):t2R}is fulfilled. Proposition 1.6. If the process Uon Xis pullback b D0-asymptotically compact and closed, then the family of sets {⇤(b D0,t); t2R}is invariant for U, that is ⇤(b D0,t)=U(t, ⌧)⇤(b D0,⌧)8⌧t. Proof. Consider fixed ⌧<tand y2⇤(b D0,⌧).Then, from the sequential characterization of ⇤(b D0,⌧), there exist sequences {⌧n}⇢(1,⌧]and{xn}⇢Xwith ⌧n!1and xn2D0(⌧n)foralln,suchthatU(⌧,⌧n)xn!y. From this, taking into account that U(t, ⌧n)=U(t, ⌧)U(⌧,⌧n)foralln,andtheprocessUis closed and pullback b D0-asymptotically compact, it holds that U(t, ⌧)y2⇤(b D0,t). Thus, U(t, ⌧)⇤(b D0,⌧)⇢⇤(b D0,t). Thereupon, we will prove that ⇤(b D0,t)⇢U(t, ⌧)⇤(b D0,⌧). To do this, consider y2⇤(b D0,t)fixed.Then,thereexistssequences{⌧n}⇢(1,t]and{xn}⇢Xwith ⌧n!1and xn2D0(⌧n)foralln,suchthat lim n!1 dX(U(t, ⌧n)xn,y)=0.(1.9) Since ⌧n!1,thereexistsn(⌧)1 such that ⌧n⌧for all nn(⌧).Therefore, we have U(t, ⌧n)xn=U(t, ⌧)U(⌧,⌧n)xn8nn(⌧).(1.10) Now, since the process Uon Xis pullback b D0-asymptotically compact, there exists subsequences {⌧n}nn(⌧)and {xn}nn(⌧)(relabeled the same), such that U(⌧,⌧n)xn! z2⇤(b D0,⌧).From this, taking into account that the process Uis closed, (1.9) and (1.10), it satisfies that U(t, ⌧)z=y.Therefore,⇤( b D0,t)⇢U(t, ⌧)⇤(b D0,⌧). Chapter 1. Theory of Pullback Attractors 39 In what follows, consider a nonempty class Dof families parameterized in time b D={D(t):t2R}⇢P(X).The class Dis called a universe in P(X). Then, we have the following definition. Definition 1.7. The family b D0={D0(t):t2R}⇢P(X)is pullback D-absorbing for the process Uon Xif for any t2Rand any b D2D, there exists ⌧0(b D,t)<t such that U(t, ⌧)D(⌧)⇢D0(t)for all ⌧⌧0(b D,t). Observe that in the above definition, b D0does not necessarily belong to the class D. Proposition 1.8. If the family b D0is pullback D-absorbing for the process Uon X, then ⇤(b D,t)⇢⇤(b D0,t)for all b D2Dand t2R. Moreover, if b D02D,then ⇤(b D0,t)⇢D0(t)Xfor all t2R. Proof. Consider fixed b D2Dand t2R.If ⇤(b D,t) is nonempty, then given y2⇤(b D,t), from the sequential characterization of ⇤(b D,t), there exist sequences {⌧n}⇢(1,t]and{xn}⇢Xwith ⌧n!1and xn2D(⌧n)foralln,suchthat U(t, ⌧n)xn!y. On the other hand, since the family b D0is pullback D-absorbing for the process U,thereexists⌧0(b D,t)tsuch that U(t, ⌧)D(⌧)⇢D0(t)forall ⌧⌧0(b D,t).Then, we deduce that there exists n(⌧0)1 such that U(t, ⌧n)xn2 D0(t)forallnn(⌧0).Now, consider a subsequence of {⌧n}which satisfies that ⌧nktkand ynk:= U(tk,⌧nk)xnk2D0(tk)forallk1.From this, taking into account that U(t, t k)ynk=U(t, ⌧nk)xnkfor all k1, it fulfils that y2⇤(b D0,t).Therefore, ⇤(b D,t)⇢⇤(b D0,t)forall b D2Dand t2R. Finally, if b D02D,we will prove that ⇤(b D0,t)⇢D0(t)Xfor all t2R. Consider t2Rfixed. If ⇤(b D0,t) is nonempty, given y2⇤(b D0,t), there exist {⌧n}⇢(1,t]and{xn}⇢Xwith ⌧n!1and xn2D0(⌧n), such that U(t, ⌧n)xn!y. Now, taking into account that the family b D0is pullback D- absorbing, there exists n02Nsuch that U(t, ⌧n)D0(⌧n)⇢D0(t)forallnn0. Therefore, y2D0(t)X. Now, we have the following definition. Definition 1.9. A process Uon Xis said to be pullback D-asymptotically compact if it is pullback b D-asymptotically compact for any b D2D(cf. Definition 1.3). Proposition 1.10. If b D0={D0(t):t2R}⇢P(X)is a pullback D-absorbing family for the process Uon Xand the process Uis pullback b D0-asymptotically compact, then the process Uis also pullback D-asymptotically compact. Proof. Consider fixed t2R,b D2D,andthesequences{⌧n}⇢(1,t]and{xn}⇢ Xsuch that ⌧n!1and xn2D(⌧n)foralln. Our aim is to prove that the sequence {U(t, ⌧n)xn}is relatively compact in X. Since the family b D0is pullback D-absorbing for the process Uon X,forany k1, there exist ⌧nk2{⌧n}such that ⌧nktkand ynk:= U(tk,⌧nk)xnk2 40 1.2. Existence and relationships between pullback attractors D0(tk).Now, taking into account that Uis pullback b D0-asymptotically compact, it satisfies that the sequence {U(t, t k)ynk}is relatively compact. Finally, since U(t, t k)ynk=U(t, ⌧nk)xnk,thenwehaveprovedthatthesequence{U(t, ⌧nk)xnk} is relatively compact in X. 1.2 Existence and relationships between pullback attractors Definition 1.11. Consider a family AD={AD(t):t2R}⇢P(X). Then, it is called the minimal pullback D-attractor for the process Uif the following properties are satisfied: (a) the set AD(t)is a nonempty compact subset of Xfor any t2R, (b) ADis pullback D-attracting, i.e. lim ⌧!1 distX(U(t, ⌧)D(⌧),AD(t)) = 0 8t2R8b D2D, (c) ADis invariant, i.e. U(t, ⌧)AD(⌧)=AD(t)for all ⌧t, (d) ADis minimal, i.e. if b C={C(t):t2R}⇢P(X)is a family of closed sets which is pullback D-attracting, then AD(t)⇢C(t)for all t2R. The uniqueness of the minimal pullback D-attractor comes from its own definition (cf. (d)). See also Remark 1.14 (i). Now, to prove the main result of this section, we need the following proposition, which is a consequence of Propositions 1.5 and 1.6. Proposition 1.12. Suppose that the process Uon Xis pullback D-asymptotically compact and closed. Then, for any b D2Dand t2R, the set ⇤(b D,t)is a nonempty compact subset of X, which is invariant for Uand attracts b Din the pullback sense. In addition, for each b D2D,the family {⇤(b D,t):t2R}is minimal amongst all the pullback attracting families of closed sets. The following theorem guarantees the existence of the minimal pullback attractor. Theorem 1.13. Assume that U:R2 d⇥X!Xis a closed process, Dis a universe in P(X)and b D0={D0(t):t2R}⇢P(X)is a pullback D-absorbing family for U. Moreover, suppose that Uis pullback b D0-asymptotically compact. Then, the family AD={AD(t):t2R},which is given by AD(t)= [ b D2D ⇤(b D,t) X 8t2R,(1.11) is the minimal pullback D-attractor for the process U. In addition, if b D02D, then AD(t)⇢D0(t)Xfor all t2R. Chapter 1. Theory of Pullback Attractors 41 Proof. Firstly, we will prove that AD(t) is a nonempty compact set and fulfils AD(t)⇢⇤(b D0,t)forallt2R. Consider t2Rfixed. Making use of Propositions 1.10 and 1.12, and taking into account (1.11), AD(t)isanonemptyclosed set. In fact, thanks to Propositions 1.5 and 1.8, we have that AD(t) is compact and AD(t)⇢⇤(b D0,t). Moreover, the family ADis pullback D-attracting. Namely, from (1.11) we deduce distX(U(t, ⌧)D(⌧),AD(t)) distX(U(t, ⌧)D(⌧),⇤(b D,t)), for all b D2Dand (t, ⌧)2R2 d.From this, taking into account Proposition 1.12, it holds that the family ADis pullback D-attracting. Thereupon, the invariance of ADwill be proved. Firstly, we will show that AD(t)⇢U(t, ⌧)AD(⌧) for all (t, ⌧)2R2 d.Consider fixed (t, ⌧)2R2 dand y2A D(t). From (1.11), we deduce that there exist two sequences {b Dn}⇢Dand {yn}⇢Xsuch that yn2⇤(b Dn,t)forallnand yn!y. Since ⇤(b D,t) is invariant (cf. Proposition 1.6), there exists a sequence {xn}⇢Xwith xn2⇤(b Dn,⌧)⇢A D(⌧)suchthat yn=U(t, ⌧)xnfor all n.SinceAD(⌧)isacompactsetandtheprocessUis closed, y2U(t, ⌧)AD(⌧).Now, to prove that U(t, ⌧)AD(⌧)⇢A D(t), we fix (t, ⌧)2R2 dand y2A D(⌧).We will check that U(t, ⌧)y2A D(t).Since y2A D(⌧), there exist two sequences {b Dn}⇢Dand {yn}⇢Xsuch that yn2⇤(b Dn,⌧)andyn!y.From this, taking into account that ⇤(b Dn,t)=U(t, ⌧)⇤(b Dn,⌧)andAD(t) is compact, it holds that the sequence {xn}⇢X,givenbyxn:= U(t, ⌧)yn2A D(t)foralln,is relatively compact in X. Finally, bearing in mind that the process Uis closed, we conclude that U(t, ⌧)y2A D(t). The minimality of ADis due to Proposition 1.12 and (1.11). Namely, assume that b C={C(t):t2R}⇢P(X)isafamilyofclosedsetssuchthatforany b D={D(t):t2R}2D, lim ⌧!1 distX(U(t, ⌧)D(⌧),C(t)) = 0. Consider fixed t2Rand y2A D(t).From (1.11), we deduce that there exists two sequences {b Dn}⇢Dand {yn}⇢Xsuch that yn2⇤(b Dn,t)forallnand yn!y. In fact, {yn}⇢C(t) (cf. Proposition 1.12). Then, y2C(t)andtherefore, AD(t)⇢C(t). Finally, when b D02D, it holds that AD(t)⇢D0(t)Xfor all t2R, thanks to (1.11) and Proposition 1.8. Remark 1.14. (i) If AD2D, then it is the unique family of closed subsets in D that satisfies (b) and (c) in Definition 1.11. A sufficient condition for AD2D is to have that b D02D, the set D0(t)is closed for all t2Rand the universe D is inclusion-closed, that means that if b D2Dand b D0={D0(t):t2R}⇢P(X) satisfies that D0(t)⇢D(t)for all t2R, then b D02D. (ii) The universe of fixed nonempty bounded subsets of Xis denoted by DX F. Then, the corresponding minimal pullback DX F-attractor for the process Uis the attractor defined by Crauel, Debussche and Flandoli (cf. [55, Theorem 1.1]). 48 2.1. Statement of the problem. Existence results for any fixed b2R+and x=(x1,...,x n)7! z(x)=a(l( n X i=1 xiwi)).(2.11) In what follows, for simplicity we denote ⇠=((u⌧,w 1),...,(u⌧,w n)). Firstly, we are going to prove that gis a Caratheodory function. Consider xfixed. The function g(·,x)ismeasurablebecause gj(·,x)=jz(x)xj+(f( n X i=1 xiwi),w j)+hh(·),w ji is measurable function as a consequence of Fubini’s Theorem. Secondly, we need to check that the function g(t, ·) is continuous a.e. t2[⌧,T]. Indeed gj(t, x)=jz(x)xj+(f( n X i=1 xiwi),w j)+hh(t),w ji is a continuous function with respect to the second variable, because the functions zand x2Rn7! (f(Pn i=1 xiwi),w j) are continuous. Now we are going to prove that there exists a function m2L1(⌧,T)suchthat |g(t, x)|m(t)8(t, x)2R. From the definition of R, we deduce |x|b+|⇠|=: CR, |(f( n X i=1 xiwi),w j)|21/2Cf|⌦|1/2|wj|2+2 1/2CfCR( n X i=1 |wi|2)|wj|2. Observe that taking into account (2.11) and making use of the continuity of the function ain the compact interval I0:= [|l|2CRPn i=1 |wi|2,|l|2CRPn i=1 |wi|2], there exists M>0 such that z(x)M8x2Rn:|x|CR. Then, bearing this in mind, we deduce |gj(t, x)| jz(x)|xj|+|(f( n X i=1 xiwi),w j)|+|hh(t),w ji| jMCR+2 1/2Cf|⌦|1/2|wj|2+2 1/2CfCR( n X i=1 |wi|2)|wj|2+|hh(t),w ji|. Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 49 Therefore, m= n X j=1 jMCR+ n X j=1 21/2Cf[|⌦|1/2+CR( n X i=1 |wi|2)]|wj|2 + n X j=1 |hh(·),w ji|2L1(⌧,T). In conclusion, there exists a local solution to (2.10). Step 2. Uniqueness of local solution. Since the function ais locally Lipschitz, for any bounded interval [R, R]ofR,thereexistsapositiveconstant La(R)suchthat |a(x)a(y)|La(R)|xy|8x, y 2[R, R]. Assume that there exist two solutions '1 n,' 2 nof the ordinary di↵erential system (2.10) in (⌧,t1)and(⌧,t2)respectively.Then,itholds (('1 nj(t)'2 nj(t))0=gj(t, '1 n(t)) gj(t, '2 n(t)),t2(⌧,min{t1,t 2}), ('1 nj '2 nj)(⌧)=0,j=1,...,n, (2.12) where gj(t, 'i n(t)) = ja(l(ui n(t)))'i nj(t)+(f(ui n(t)),w j)+hh(t),w jifor i=1,2. Then, multiplying (2.12) by '1 nj '2 nj,summing from j=1tonand making use of (2.5), we obtain 1 2 d dt|u1 n(t)u2 n(t)|2 2+a(l(u1 n(t)))ku1 n(t)u2 n(t)k2 2 |a(l(u1 n(t))) a(l(u2 n(t)))||((u2 n(t),u 1 n(t)u2 n(t)))|+⌘|u1 n(t)u2 n(t)|2 2. Since u1 n,u 2 n2C([⌧,min{t1,t 2}]; L2(⌦)),it fulfils that u1 n(t),u 2 n(t)2Sfor all t2 [⌧,min{t1,t 2}],where Sis a bounded set of L2(⌦).In addition, taking into account that l2L2(⌦), it satisfies that {l(ui n(t))}t2[⌧,min{t1,t2}]2[R, R]fori=1,2,for some R>0.Hence, using (2.3), (2.5) and the fact that the function ais locally Lipschitz (with Liptschitz constant La(·)), we deduce d dt|u1 n(t)u2 n(t)|2 2C|u1 n(t)u2 n(t)|2 2, where C=(La(R))2|l|2 2nC2 R+4m⌘ 2m. Then, using the Gronwall Lemma, we have |u1 n(t)u2 n(t)|2 2|u1 n(⌧)u2 n(⌧)|2 2eC(t⌧). 50 2.1. Statement of the problem. Existence results Now, we will show the existence and uniqueness of weak solutions and the continuity of the solution in L2(⌦) with respect to the initial data. Theorem 2.4. Suppose that the function ais locally Lipschitz and fulfils (2.3), f2C(R)satisfies (2.4) and (2.5), h2L2 loc(R;H1(⌦)) and l2L2(⌦).Then, for each initial datum u⌧2L2(⌦), there exists a weak solution to the problem (2.2). In addition, this solution behaves continuously in L2(⌦)w.r.t. the initial data. Proof. We split the proof into two steps. Step 1. Existence of weak solution. Multiplying by 'nj(t) in (2.9), summing from j=1tonand using (2.3), we obtain d dt|un(t)|2 2+2mkun(t)k2 22(f(un(t)),u n(t)) + 2hh(t),u n(t)ia.e. t2(⌧,tn), where (⌧,tn) is the interval of existence of maximal solution. By the Cauchy inequality (cf. [57, Appendix B, p. 622]) and (2.4), d dt|un(t)|2 2+mkun(t)k2 24C2 f|⌦| 1m+4C2 f 1m|un(t)|2 2+2 mkh(t)k2 ⇤a.e. t2(⌧,tn), where 1is the first eigenvalue of with zero Dirichlet boundary conditions. Integrating between ⌧and twith ⌧<t<t n, we obtain |un(t)|2 2+mZt ⌧ kun(s)k2 2ds |u⌧|2 2+4C2 f|⌦|(T⌧) 1m+4C2 f 1mZT ⌧ |un(s)|2 2ds +2 mZT ⌧ kh(s)k2 ⇤ds. Therefore, the Gronwall lemma implies that {un}is well defined for all time t⌧, and actually bounded in L1(⌧,T;L2(⌦))\L2(⌧,T;H1 0(⌦)) for all T>⌧.Thus, taking this into account together with the fact that each un2C([⌧,T]; L2(⌦)),we deduce that there exists a positive constant C1such that |un(t)|2C18t2[⌧,T]8n1. From this, bearing in mind that a2C(R;R+)andl2L2(⌦),it satisfies that there exists a positive constant MC1>0 such that a(l(un(t))) MC18t2[⌧,T]8n1. Now, we have ZT ⌧ |a(l(un(t)))|2kun(t)k2 ⇤dt (MC1)2ZT ⌧ kun(t)k2 2dt. (2.13) Taking into account that {un}is bounded in L2(⌧,T;H1 0(⌦)), we deduce that the sequence {a(l(un))un}is bounded in L2(⌧,T;H1(⌦)). Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 51 On the other hand, using (2.4), we have ZT ⌧Z⌦ |f(un(x, t))|2dxdt ZT ⌧Z⌦ 2C2 f(1 + |un(x, t)|2)dxdt 2C2 f|⌦|(T⌧)+2C2 fZT ⌧ |un(t)|2 2dt. (2.14) Now, using that {un}is bounded in L1(⌧,T;L2(⌦)), we have that {f(un)}is bounded in L2(⌧,T;L2(⌦)). To prove that the sequence {u0 n}is bounded in L2(⌧,T;H1(⌦)), we need first to define the following projectors: e Pn:H1(⌦)! H1(⌦) f7! [2H1 0(⌦)7! h e Pnf,i:= hf,Pni], where Pn:L2(⌦)! Vn:= span[w1,...,w n] 7! Pn j=1(, wj)wj. Observe that e Pnis the continuous extension in H1(⌦) of Pn.Then,inwhatfollows, we will make an abuse of notation and denote this projection by Pn. Bearing in mind (2.13), (2.14) and the definitions of the above projectors, we have that the sequence {u0 n}is bounded in L2(⌧,T;H1(⌦)), since ZT ⌧ ku0 n(t)k2 ⇤dt =ZT ⌧ ka(l(un(t)))un(t)+Pnf(un(t)) + Pnh(t)k2 ⇤dt 3(MC1)2ZT ⌧ kun(t)k2 2dt +3 1ZT ⌧ |Pnf(un(t))|2 2dt +3ZT ⌧ kPnh(t)k2 ⇤dt (3(MC1)2+6C2 f1 1)ZT ⌧ kun(t)k2 2dt +6C2 f1 1|⌦|(T⌧)+3ZT ⌧ kh(t)k2 ⇤dt. Therefore, making use of compactness arguments and the Aubin-Lions lemma, there exist a subsequence of {un}(relabeled the same) and u2L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦)) with u02L2(⌧,T;H1(⌦)),such that 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : un ⇤ *uweakly-star in L1(⌧,T;L2(⌦)), un*uweakly in L2(⌧,T;H1 0(⌦)), u0 n*u 0weakly in L2(⌧,T;H1(⌦)), un!ustrongly in L2(⌧,T;L2(⌦)), un(x, t)!u(x, t)a.e.(x, t)2⌦⇥(⌧,T), un(t)!u(t) strongly in L2(⌦) a.e. t2(⌧,T), f(un)*⇠ 1weakly in L2(⌧,T;L2(⌦)), a(l(un))un*⇠ 2weakly in L2(⌧,T;H1 0(⌦)), (2.15) 52 2.1. Statement of the problem. Existence results for all T>⌧. Now, we need to check that ⇠1=f(u)and⇠2=a(l(u))u.Sinceunconverges to ustrongly in L2(⌦), we deduce that un(x, t)!u(x, t)8(x, t)2⌦⇥(⌧,T)\N1,(2.16) un(t)!u(t) strongly in L2(⌦)8t2(⌧,T)\N2,(2.17) where N1is a null set in RN+1 and N2is a null set in R. From this, we can deduce that ⇠1=f(u) applying [85, Lemme 1.3, p. 12], since f2C(R), {f(un)}is bounded in L2(⌧,T;L2(⌦)) and converges pointwisely to f(u) a.e. ⌦⇥(⌧,T) making use of (2.16). Finally, we will prove that ⇠2=a(l(u))u. As a2C(R;R+), l2L2(⌦) and (2.17) holds, it satisfies a(l(un(t))) !a(l(u(t))) 8t2(⌧,T)\N2. Therefore, a(l(un(t)))un(x, t)!a(l(u(t)))u(x, t)8(x, t)2⌦⇥(⌧,T)\(N1[(⌦⇥N2)), where N1[(⌦⇥N2)isanullsetinRN+1.In addition, {a(l(un))un}is bounded in L2(⌧,T;H1 0(⌦)). Then, applying again [85, Lemme 1.3, p. 12], ⇠2=a(l(u))u follows. Now, passing to the limit in (2.9), taking into account (2.15) together with the fact that [n2NVnis dense in H1 0(⌦), (2.6) holds for all v2H1 0(⌦).Therefore, to prove that uis a weak solution to (2.2), we only need to check that u(⌧)=u⌧, which makes complete sense since u2C([⌧,T]; L2(⌦)). On the one hand, consider fixed n,'2H1(⌧,T)with'(T)=0and'(⌧)6=0, and w2Vn.Multiplying by 'in (2.9), integrating between ⌧and T,weobtainfor all µ>n (u⌧,w)'(⌧)ZT ⌧ '0(t)(uµ(t),w)dt +ZT ⌧ a(l(uµ(t)))huµ(t),wi'(t)dt =ZT ⌧ (f(uµ(t)),w)'(t)dt +ZT ⌧ hh(t),wi'(t)dt. (2.18) Taking limit when µ!1and using (2.15), we deduce (u⌧,w)'(⌧)ZT ⌧ '0(t)(u(t),w)dt +ZT ⌧ a(l(u(t)))hu(t),wi'(t)dt =ZT ⌧ (f(u(t)),w)'(t)dt +ZT ⌧ hh(t),wi'(t)dt. (2.19) On the other hand, multiplying by 'in (2.6) and integrating between ⌧and T,we obtain (u(⌧),w)'(⌧)ZT ⌧ '0(t)(u(t),w)dt +ZT ⌧ a(l(u(t)))hu(t),wi'(t)dt =ZT ⌧ (f(u(t)),w)'(t)dt +ZT ⌧ hh(t),wi'(t)dt. Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 53 Comparing (2.19) with the above expression, we have (u(⌧),w)'(⌧)=(u⌧,w)'(⌧). As '(⌧)6=0and{wj}is a Hilbert basis of L2(⌦), we conclude u(⌧)=u⌧. Step 2. Uniqueness of solution and continuity w.r.t. initial data. Assume that there exist two weak solutions, u1(·;⌧,u1 ⌧)andu2(·;⌧,u2 ⌧), to (2.2). For short, we will denote ui(·)=ui(·;⌧,ui ⌧)fori=1,2. From the energy equality, we obtain 1 2 d dt|u1(t)u2(t)|2 2+a(l(u1(t)))ku1(t)u2(t)k2 2 =[a(l(u2(t)))a(l(u1(t)))]((u2(t),u 1(t)u2(t)))+(f(u1(t))f(u2(t)),u 1(t)u2(t)) a.e. t2[⌧,T]. Since u1,u 22C([⌧,T]; L2(⌦)),there exists a bounded set S⇢L2(⌦) such that {ui(t)}t2[⌧,T]⇢Sfor i=1,2. Besides, taking into account that l2L2(⌦), there exists a constant R>0 such that {l(ui(t))}t2[⌧,T]⇢[R, R]fori=1,2. Then, making use of (2.3), (2.5) and the locally Lipschitz continuity of the function a, we obtain 1 2 d dt|u1(t)u2(t)|2 2+mku1(t)u2(t)k2 2 La(R)|l|2|u1(t)u2(t)|2ku2(t)k2ku1(t)u2(t)k2+⌘|u1(t)u2(t)|2 2, where La(R) denotes the Lipschitz constant of the function ain [R, R]. Now, applying the Cauchy inequality to the above expression, we have d dt|u1(t)u2(t)|2 2C(t)|u1(t)u2(t)|2 2a.e. t2(⌧,T) where C(t)=(La(R))2|l|2 2ku2(t)k2 2+2m⌘ m. Thus, we deduce |u1(t)u2(t)|2 2|u1 ⌧u2 ⌧|2 2eRt ⌧C(s)ds 8t2[⌧,T]. Both results, the uniqueness of solution to (2.2) and the continuity w.r.t. the initial data, follow immediately. In the following result, we will study the regularising e↵ect of the equation. In addition, taking a more regular initial datum, the existence of a strong solution will be analysed. Theorem 2.5. Under the assumptions of Theorem 2.4, if h2L2 loc(R;L2(⌦)), for every ">0and T>⌧+", the weak solution ubelongs to C((⌧,T],H1 0(⌦))\ L2(⌧+", T;D()).In fact, if the initial condition u⌧2H1 0(⌦), then the function u2C([⌧,T],H1 0(⌦)) \L2(⌧,T;D()) for every T>⌧. 54 2.1. Statement of the problem. Existence results Proof. We split the proof into two steps. Step 1. Regularising e↵ect. Multiplying by j'nj(t) in (2.9), summing from j=1ton, and using (2.3), (2.4) and the Cauchy and Poincar´e inequalities, it yields d dtkun(t)k2 2+m|un(t)|2 24C2 f|⌦| m+4C2 f 1mkun(t)k2 2+2 m|h(t)|2 2a.e. t⌧. Integrating between sand twith ⌧<stT, we obtain kun(t)k2 2+mZt s |un(r)|2 2dr 4C2 f|⌦|(T⌧) m+4C2 f 1mZT ⌧ kun(r)k2 2dr +2 mZT ⌧ |h(r)|2 2dr +kun(s)k2 2.(2.20) Now, integrating in sbetween ⌧and t,wededuce (t⌧)kun(t)k2 24C2 f|⌦|(T⌧)2 m+✓4C2 f(T⌧) 1m+1 ◆ZT ⌧ kun(r)k2 2dr +2(T⌧) mZT ⌧ |h(r)|2 2dr. Therefore, kun(t)k2 24C2 f|⌦|(T⌧)2 "m+✓4C2 f(T⌧)+1m "1m◆ZT ⌧ kun(r)k2 2dr +2(T⌧) "mZT ⌧ |h(r)|2 2dr, for all t2["+⌧,T]with"2(0,T ⌧). From this and taking into account the boundedness of {un}in L2(⌧,T;H1 0(⌦)) (cf. Theorem 2.4), we deduce that {un}is bounded in L1("+⌧,T;H1 0(⌦)).As a byproduct, the boundedness of {un}in L2(⌧+", T;D()) is immediate just taking s="and t=Tin (2.20). In addition, making use of this more regular boundedness, we deduce that the sequence {u0 n}is bounded in L2(⌧+", T ;L2(⌦)). Thanks to the uniqueness of the weak solution, unconverge to uweakly in L2(⌧+ ", T;D()) and u0 nconverge to u0weakly in L2(⌧+", T;L2(⌦)).As a consequence, u2L2(⌧+", T;D()) \C((⌧,T]; H1 0(⌦)). Step 2. Strong solution. In this step if u⌧2H1 0(⌦), we will show that u2L2(⌧,T;D()) \C([⌧,T]; H1 0(⌦)) for all T>⌧.To that end, we multiply by j'nj in (2.9), sum from j=1tonand use (2.3), obtaining 1 2 d dtkun(t)k2 2+m|un(t)|2 2(f(un(t)),un(t)) + (h(t),un(t)) (2.21) a.e. t2(⌧,T). Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 55 Now, from (2.4) and the Cauchy inequality, we deduce (f(un(t)),un(t)) 1 m|f(un(t))|2 2+m 4|un(t)|2 2 2C2 f|⌦| m+2C2 f m|un(t)|2 2+m 4|un(t)|2 2, (h(t),un(t)) 1 m|h(t)|2 2+m 4|un(t)|2 2. Taking this into account, from (2.21), we deduce that d dtkun(t)k2 2+m|un(t)|2 24C2 f 1mkun(t)k2 2+4C2 f|⌦| m+2 m|h(t)|2 2 a.e. t>⌧. Integrating between ⌧and t2[⌧,T], we have kun(t)k2 2+mZt ⌧ |un(s)|2 2ds ku⌧k2 2+4C2 f 1mZT ⌧ kun(s)k2 2ds +4C2 f|⌦|(T⌧) m+2 mZT ⌧ |h(s)|2 2ds. Taking into account that {un}is bounded in L2(⌧,T;H1 0(⌦)) (cf. Theorem 2.4), we deduce that {un}is bounded in L1(⌧,T;H1 0(⌦)) \L2(⌧,T;D()).As a result, the sequence {u0 n}is bounded in L2(⌧,T;L2(⌦)).Then, thanks to the uniqueness of a weak solution, it holds that unconverge to uweakly-star in L1(⌧,T;H1 0(⌦)) and weakly in L2(⌧,T;D()), and u0 nconverge to u0weakly in L2(⌧,T;L2(⌦)). Therefore, since u2L1(⌧,T;H1 0(⌦)) \L2(⌧,T;D()) and u02L2(⌧,T;L2(⌦)), we obtain that u2C([⌧,T]; H1 0(⌦)). 2.2 Analysis of the stationary problem In this section we study the elliptic problem ⇢a(l(u))u=f(u)+hin ⌦, u=0 on@⌦,(2.22) where the functions aand fare globally Lipschitz, with respective Lipschitz constants La,L f0 and there exists a positive constant M>0 such that 0<ma(s)M8s2R.(2.23) In addition, his time-independent, i.e. h2H1(⌦). We analyse the existence of solutions to (2.22) making use of a corollary of the Brouwer fixed point theorem. The uniqueness as well as the global exponential stability are also studied under suitable assumptions. 56 2.2. Analysis of the stationary problem Definition 2.6. A solution to (2.22) is a function u⇤2H1 0(⌦)such that a(l(u⇤))((u⇤,v)) = (f(u⇤),v)+hh, vi8v2H1 0(⌦). In the following result we analyse the existence, uniqueness and regularity of the stationary solutions of the problem (2.2) (the idea of the proof is close to that in [91]). Theorem 2.7. Assume that aand fare globally Lipschitz functions, with Lipschitz constants Laand Lfrespectively, (2.23) is satisfied, h2H1(⌦),l2L2(⌦)and m> 1 1Lf.Then: 1. There exists at least one solution to (2.22). In addition, any solution u⇤to (2.22) fulfils ku⇤k2⌥:=1/2 1|⌦|1/2|f(0)|+khk⇤ m1 1Lf .(2.24) Further, if h2L2(⌦), then the solutions given above belong in fact to D(). 2. Under the additional assumption 1/2 1|l|2La⌥<m1 1Lf,(2.25) problem (2.22) possesses a unique solution. Proof. We split the proof into four steps. Step 1. Existence. Let us consider the orthonormal Hilbert basis {wj:j1} of L2(⌦) consisting of the eigenvectors associated with eigenvalues {j:j1}of the operator with zero Dirichlet boundary condition in ⌦. Foreachn1,let us denote Vn= span[w1,...,w n],with the inner product ((·,·)) and norm k·k 2. Now, the operators Rn:Vn!Vnfor all n1 are defined as follows ((Rnu, v)) = ha(l(u))u, vi(f(u),v)hh, vi8u, v 2Vn. Observe that each Rnu2Vnis well defined thanks to the Riesz Theorem, since the right hand side is a continuous linear map from Vnto R. In addition, Rnis continuous. Namely, making use of (2.23), the Poincar´e inequality and the Lipschitz continuity of the functions aand f,wededuce ((RnuRn˜u, v)) =ha(l(u))u+a(l(˜u))˜uf(u)+f(˜u),vi =ha(l(u))((u˜u)+(a(l(˜u)) a(l(u)))˜u, vi+(f(˜u)f(u),v) (M+La|l|21/2 1k˜uk2+Lf1 1)k˜uuk2kvk2, for all u, ˜u, v 2Vn.Therefore, kRnuRn˜uk2(M+La|l|21/2 1k˜uk2+Lf1 1)k˜uuk2. Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 57 for all u, ˜u2Vn. On the other hand, making use again of (2.23), the Poincar´e inequality and the global Liptchiz continuity of f, we have ((Rnu, u)) = ha(l(u))u, ui(f(u),u)hh, ui =ha(l(u))u, ui(f(u)f(0),u)(f(0),u)hh, ui mkuk2 2Lf1 1kuk2 2|f(0)||⌦|1/21/2 1kuk2khk⇤kuk2, for all u2Vn. Therefore, taking ⌥:=1/2 1|⌦|1/2|f(0)|+khk⇤ m1 1Lf , we obtain ((Rnu, u)) 08u2Vn/kuk2=⌥. Now, making use of a corollary of the Brouwer fixed point theorem (see [85, Lemme 4.3, p.53]), we deduce that for each n1 there exists un2Vnsuch that Rn(un)=0,with kunk2⌥.(2.26) Therefore, it verifies ha(l(un))un,vi=(f(un),v)+hh, vi8v2Vn.(2.27) Now, using the boundedness of {un}in H1 0(⌦)by⌥and the compact embedding H1 0(⌦),!L2(⌦), we can extract a subsequence {unk}⇢{un}which fulfils (unk*u ⇤weakly in H1 0(⌦), unk!u⇤strongly in L2(⌦), where u⇤2H1 0(⌦) is a solution to (2.22). To check that, just take limit in (2.27) and make use of the assumptions made on a,land f. In addition, observe that u⇤ fulfils (2.26). Step 2. The a priori estimate (2.24). So far, we only have proven that there exists at least a solution to problem (2.22) and u⇤verifies (2.24). But this does not imply that any u⇤solution to problem (2.22) fulfils (2.24) since the uniqueness of solution is not guaranteed. Therefore, let us prove that any u⇤solution to problem (2.22) verifies (2.24). Consider fixed u⇤a solution to (2.22). It holds mku⇤k2 2|f(u⇤)f(0)|2|u⇤|2+|f(0)||⌦|1/2|u⇤|2+khk⇤ku⇤k2. Using that the function fis globally Lipschitz and the Poincar´e inequality, we obtain mku⇤k2 2Lf1 1ku⇤k2 2+|f(0)||⌦|1/21/2 1ku⇤k2+khk⇤ku⇤k2. Therefore, u⇤satisfies (2.24). 64 2.3. Minimal pullback attractors in L2-norm {a(l(un))un}nn1and {(un)0}nn1are bounded in L2(t2,t;H1(⌦)).Then, using the Aubin-Lions lemma, there exists u2L1(t2,t;L2(⌦))\L2(t2,t;H1 0(⌦)) with u02L2(t2,t;H1(⌦)), such that for a subsequence (relabeled the same) it holds 8 > > > > > > > > > > > < > > > > > > > > > > > : un⇤ *uweakly-star in L1(t2,t;L2(⌦)), un*uweakly in L2(t2,t;H1 0(⌦)), (un)0*u 0weakly in L2(t2,t;H1(⌦)), un!ustrongly in L2(t2,t;L2(⌦)), un(s)!u(s) strongly in L2(⌦) a.e. s2(t2,t), f(un)*f(u)weaklyinL2(t2,t;L2(⌦)), a(l(un))un*a(l(u))uweakly in L2(t2,t;H1(⌦)). (2.38) Observe that the last two convergences in (2.38) have been obtained arguing in the same way as in the proof of Theorem 2.4, making use of [85, Lemme 1.3, p. 12]. Furthermore, u2C([t2,t]; L2(⌦)) and using (2.38), it is not difficult to prove that ufulfils (2.6) in the interval (t2,t). Since {(un)0}nn1is bounded in L2(t2,t;H1(⌦)), we have that {un}nn1is equicontinuous in H1(⌦) on [t2,t].Namely, fixed ">0, consider s1,s 22[t2,t] with |s2s1|<1, then kun(s2)un(s1)k2 ⇤ sup v2H1 0(⌦)/kvk2=1 hZs2 s1 (un(r))0dr, vi!2 ✓Zs2 s1 k(un(r))0k⇤dr◆2 ⇢3(t)|s2s1|. Then, it just simply takes "=min{"2/⇢3(t),1}. In addition, as {un}nn1is bounded in C([t2,t]; L2(⌦)) and the embedding L2(⌦)⇢H1(⌦) is compact, by the Arzela- Ascoli theorem, we obtain (for another subsequence, relabeled again the same) un!ustrongly in C([t2,t]; H1(⌦)).(2.39) Now, consider a sequence {sn}⇢[t2,t]whichconvergestos⇤.Since{un}nn1is bounded in C([t2,t]; L2(⌦)),there exist a subsequence of {un(sn)}nn1(relabeled the same) and v2L2(⌦) such that un(sn)*vweakly in L2(⌦). (2.40) Let us prove that v=u(s⇤). Fixed ">0,from (2.39) we deduce that there exists n"2Nsuch that kun(s)u(s)k⇤" 28nn✏8s2[t2,t]. From this and using that the function u2C([t2,t]; H1(⌦)),we deduce un(sn)!u(s⇤) strongly in H1(⌦).(2.41) Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 65 Then, from (2.40) and (2.41), by the uniqueness of the limit we obtain un(sn)*u(s⇤)weaklyinL2(⌦). (2.42) Observe that if we prove un!ustrongly in C([t1,t]; L2(⌦)),(2.43) in particular the sequence {u(t;⌧n,u ⌧n)}is relatively compact in L2(⌦). We establish (2.43) by contradiction. We suppose that there exist ">0,a sequence {tn}⇢[t1,t],without loss of generality converging to some t⇤,with |un(tn)u(t⇤)|2"8n1.(2.44) On the other hand, using the energy equality (2.7), the Cauchy inequality, (2.3) and (2.30), the estimate |z(s)|2 2|z(r)|2 2+2|⌦|(sr)+ 1 2(m↵1 1)Zs r kh(⇠)k2 ⇤d⇠8t2rst holds with zreplaced by uor any un. Now we define the functions Jn(s)=|un(s)|2 22|⌦|s1 2(m↵1 1)Zs t2 kh(r)k2 ⇤dr, J(s)=|u(s)|2 22|⌦|s1 2(m↵1 1)Zs t2 kh(r)k2 ⇤dr. From the regularity of uand all un, and the above inequality it holds that these functions Jand Jnare continuous and non-increasing on [t2,t].In addition, observe that using (2.38), we have Jn(s)!J(s)a.e.s2(t2,t). Hence, there exists a sequence {˜ tk}⇢(t2,t ⇤)suchthat˜ tk!t⇤when k!1and lim n!1 Jn(˜ tk)=J(˜ tk)8k1. Fix an arbitrary value ✏>0.From the continuity of Jon [t2,t],there exists k(✏)1 such that |J(˜ tk)J(t⇤)|✏/28kk(✏). Now consider n(✏)1 such that tn˜ tk(✏)and |Jn(˜ tk(✏))J(˜ tk(✏))|✏/28nn(✏). Then, since all Jnare non-increasing, we deduce Jn(tn)J(t⇤)Jn(˜ tk(✏))J(t⇤) |Jn(˜ tk(✏))J(t⇤)| |Jn(˜ tk(✏))J(˜ tk(✏))|+|J(˜ tk(✏))J(t⇤)| ✏8nn(✏). 66 2.3. Minimal pullback attractors in L2-norm As ✏>0 is arbitrary, from above we deduce lim sup n!1 Jn(tn)J(t⇤). Thus, lim sup n!1 |un(tn)|2|u(t⇤)|2. From this, (2.42) applied to the sequence {tn},itsatisfiesthatthesequence{un(tn)} converges to u(t⇤) strongly in L2(⌦),which is contradictory with (2.44). Therefore, (2.43) holds. As a consequence of the previous results, we obtain the existence of the minimal pullback attractors for the process Uon L2(⌦). Theorem 2.16. Suppose that the function ais locally Lipschitz and satisfies (2.3), f2C(R)fulfils (2.4), (2.5) and (2.30), l2L2(⌦)and h2L2 loc(R;H1(⌦)) satisfies condition (2.32) for some µ2(0,2(1m↵)). Then, there exist the minimal pullback DL2 F-attractor ADL2 Fand the minimal pullback DL2 µ-attractor ADL2 µfor the process U:R2 d⇥L2(⌦)!L2(⌦). The family ADL2 µbelongs to DL2 µand the following relationship holds ADL2 F(t)⇢A DL2 µ(t)⇢BL2(0,R 1/2 L2(t)) 8t2R.(2.45) Finally, if moreover hsatisfies sup s0✓eµs Zs 1 eµrkh(r)k2 ⇤dr◆<1,(2.46) then ADL2 F(t)=ADL2 µ(t)8t2R. Proof. The process Uis continuous on L2(⌦) and pullback DL2 µ-asymptotically compact (cf. Propositions 2.10 and 2.15 respectively). In addition, there exists a pullback DL2 µ-absorbing family (cf. Proposition 2.13) and DL2 F⇢D L2 µ.Then,from Corollary 1.15, we deduce the existence of ADL2 µand ADL2 F,as well as the first relationship appearing in (2.45). The second relation in (2.45) is straightforward making use of Theorem 1.13 and the fact that b D02D L2 µ. In addition, using that DL2 µis inclusion-closed, b D0(t)isclosedinL2(⌦) for all t2R,b D02D L2 µand the second relation in (2.45), the family ADL2 µbelongs to DL2 µ. Finally, under the assumption (2.46), the set [tTRL2(t)isboundedforeach T2R,where the expression of RL2(t) is given in the statement of Proposition 2.13. Therefore, from Corollary 1.15, we deduce that both families ADL2 Fand ADL2 µ coincide. Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 67 Remark 2.17. (i) Observe that condition (2.46) is equivalent to sup s0Zs s1 kh(r)k2 ⇤dr < 1. (ii)When h2L2 loc(R;H1(⌦)) fulfils condition (2.32) for some µ2(0,2(1m ↵)),it holds Z0 1 eskh(s)k2 ⇤ds < 182(µ, 2(1m↵)). Therefore, under the assumptions of Theorem 2.16, there exists the corresponding minimal pullback DL2 -attractor ADL2 for any 2(µ, 2(1m↵)). Furthermore, ADL2 µ(t)⇢A DL2 (t)for any t2Rfor all 2(µ, 2(1m↵)),thanks to Theorem 1.16. In fact, if hsatisfies (2.46), using the equivalence pointed out in (i), ADL2 F(t)= ADL2 µ(t)=ADL2 (t)holds for all t2Rand any 2(µ, 2(1m↵)). 2.4 Minimal attractors in H1-norm In this section we prove the existence of pullback attractors for a dynamical system associated to (2.2) in the phase space H1 0(⌦). In addition, we establish some relationships amongst these families and those analysed in Section 2.3. Observe that from Theorem 2.5, the restriction of Uto R2 d⇥H1 0(⌦) defines a process into H1 0(⌦).Since no confusion arises, we do not modify the notation and continue denoting this process as U. The following result ensures that the process Uis strong-weak continuous in H1 0(⌦). Proposition 2.18. Assume that the function ais locally Lipschitz and (2.3) holds, f2C(R)satisfies (2.4) and (2.5), and h2L2 loc(R,L 2(⌦)) and l2L2(⌦)are given. Then, the process Uis strong-weak continuous in H1 0(⌦). Proof. Consider (t, ⌧)2R2 dfixed and let {un ⌧}be a sequence of initial data which converges to u⌧strongly in H1 0(⌦).Our aim is to prove U(t, ⌧)un ⌧*U(t, ⌧)u⌧weakly in H1 0(⌦).(2.47) On the one hand, in Proposition 2.10 we have shown that the map U(t, ⌧)is continuous from L2(⌦) into itself. Therefore, U(t, ⌧)un ⌧!U(t, ⌧)u⌧strongly in L2(⌦). On the other hand, making use of (2.3), (2.4), (2.8) and the H¨older and Cauchy inequalities, we have kU(t, ⌧)un ⌧k2 2kun ⌧k2 2+2C2 f m✓Zt ⌧ (|⌦|+|U(s, ⌧)un ⌧|2 2)ds◆+1 mZt ⌧ |h(s)|2 2ds. Observe that it is not difficult to obtain a uniform estimate for {U(·,⌧)un ⌧}in L2(⌧,t;L2(⌦)) using the Gronwall lemma and (2.7). Then, the sequence {U(t, ⌧)un ⌧} is bounded in H1 0(⌦).Therefore, by the uniqueness of the limit, (2.47) holds. 68 2.4. Minimal attractors in H1-norm To prove that the process U:Rd 2⇥H1 0(⌦)!H1 0(⌦) is pullback asymptotically compact, we previously establish some uniform estimates of the solutions in a finitetime interval up to twhen the initial datum is shifted pullback far enough. To clarify the statement of the following result, we introduce the next two amounts: ⇢ext 1(t)=1+2|⌦| µ+eµ(t3) 2(m↵1 1)µ1 1Zt 1 eµ⇠kh(⇠)k2 ⇤d⇠, ⇢ext 2(t)= 1 m↵1 1✓2|⌦|+⇢ext 1(t)+ 1 m↵1 1 max r2[t2,t]Zr r1 kh(⇠)k2 ⇤d⇠◆. (2.48) [The upper script ext means that these expressions are estimates, close to those in Lemma 2.14 involving ⇢1and ⇢2, but in an extended interval, as will be indicated in the proof below.] Lemma 2.19. Assume that the function ais locally Lipschitz and (2.3) holds, f2 C(R)fulfils (2.4), (2.5) and (2.30), l2L2(⌦)and h2L2 loc(R;L2(⌦)) satisfies (2.32) for some µ2(0,2(1m↵)). Then, for any t2Rand b D2D L2 µ,there exists ⌧2(b D,t)<t3,such that for any ⌧⌧2(b D,t)and any u⌧2D(⌧),it holds 8 > > > > > > > > < > > > > > > > > : ku(r;⌧,u⌧)k2 2e⇢1(t)8r2[t2,t], Zr r1 |u(⇠;⌧,u⌧)|2 2d⇠e⇢2(t)8r2[t1,t], Zr r1 |u0(⇠;⌧,u⌧)|2 2d⇠e⇢3(t)8r2[t1,t], (2.49) where, taking into account {⇢ext i}i=1,2from (2.48), the terms {e⇢i}i=1,2,3are given by e⇢1(t)=4C2 f|⌦| m+✓1+ 4C2 f 1m◆⇢ext 2(t)+ 2 mmax r2[t2,t]Zr r1 |h(⇠)|2 2d⇠, e⇢2(t)=4C2 f|⌦| m2+1 me⇢1(t)+ 4C2 f 1m2⇢ext 2(t)+ 2 m2max r2[t1,t]Zr r1 |h(⇠)|2 2d⇠, e⇢3(t)=3(M(⇢ext 1(t),l))2e⇢2(t)+6C2 f|⌦|+6C2 f 1 ⇢ext 2(t)+3 max r2[t1,t]Zr r1 |h(⇠)|2 2d⇠, where M(⇢ext 1(t),l)is a positive constant. Proof. Analogously as in the proof of Lemma 2.14, we can obtain uniform estimates for the solutions in a longer time-interval. Actually, there exists ⌧2(b D,t)<t3 such that for any ⌧⌧2(b D,t)andanyu⌧2D(⌧), it holds |u(r;⌧,u⌧)|2 2⇢ext 1(t)8r2[t3,t], Zr r1 ku(⇠;⌧,u⌧)k2 2d⇠⇢ext 2(t)8r2[t2,t], Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 69 where {⇢ext i}i=1,2are given in (2.48). Actually, these estimates also hold for the Galerkin approximations un(·;⌧,u⌧)(forshortdenotedbyun(·)). From now on, consider fixed ⌧⌧2(b D,t)andu⌧2D(⌧). Multiplying (2.9) by j'nj(⇠),summing from j=1ton, and using (2.3), (2.4) and the Cauchy inequality, we have d d⇠kun(⇠)k2 2+m|un(⇠)|2 24C2 f|⌦| m+4C2 f 1mkun(⇠)k2 2+2 m|h(⇠)|2 2a.e. ⇠>⌧(2.50) Integrating between rand swith ⌧r1sr, we deduce kun(r)k2 2kun(s)k2 2+4C2 f|⌦| m+4C2 f 1mZr r1 kun(⇠)k2 2d⇠+2 mZr r1 |h(⇠)|2 2d⇠. Integrating the last inequality w.r.t. sbetween r1andr, kun(r)k2 2✓1+ 4C2 f 1m◆Zr r1 kun(s)k2 2ds +4C2 f|⌦| m+2 mZr r1 |h(⇠)|2 2d⇠ for all ⌧r1. Now, from the estimate on the solutions by ⇢ext 2given above, kun(r;⌧,u⌧)k2 2e⇢1(t)8r2[t2,t]8n2N,(2.51) where e⇢1(t) is given in the statement. Taking inferior limit in (2.51) and using the well-known fact that unconverge to u(·;⌧,u⌧) weakly-star in L1(t2,t;H1 0(⌦)) and u2C([t2,t]; H1 0(⌦)) (cf. Theorem 2.5), the first inequality in (2.49) holds. Now, integrating between r1andrin (2.50), we obtain in particular Zr r1 |un(⇠)|2 2d⇠ 1 mkun(r1)k2 2+4C2 f|⌦| m2+4C2 f 1m2Zr r1 kun(⇠)k2 2d⇠+2 m2Zr r1 |h(⇠)|2 2d⇠, for all ⌧r1. Therefore, Zr r1 |un(⇠)|2 2d⇠e⇢2(t)8r2[t1,t]8n2N,(2.52) where e⇢2(t) is given in the statement. Now, taking inferior limit in (2.52), bearing in mind that unconverge to u(·;⌧,u⌧)weaklyL2(r1,r;D()) for all r2[t1,t] (cf. Theorem 2.5), the second inequality in (2.49) holds. On the other hand, Zr r1 |u0 n(⇠)|2 2d⇠ 3Zr r1 a(l(un(⇠)))2|un(⇠)|2 2d⇠+3Zr r1 |f(un(⇠))|2 2d⇠+3Zr r1 |h(⇠)|2 2d⇠, 70 2.4. Minimal attractors in H1-norm for all ⌧r1. Observe that from the continuity of the function a,thefactthatl2L2(⌦) and |un(t)|2 2⇢ext 1(t)forallr2[t3,t]andn2N,wehavethatthereexitsapositive constant M(⇢ext 1(t),l)such that a(l(un(r))) M(⇢ext 1(t),l)8r2[t3,t]8n2N. Therefore, from this, (2.4) and the above estimates, it yields Zr r1 |u0 n(⇠)|2 2d⇠e⇢3(t)8r2[t1,t]8n2N, where e⇢3(t) is given in the statement. Finally, taking into account that u0 nconverge to u0(·;⌧,u⌧)weaklyinL2(r1,r;L2(⌦)) for all r2[t1,t](cf.Theorem2.5),the last inequality in (2.49) holds. Now, we introduce additional universes that involve more regularity. Definition 2.20. For each µ>0,DL2,H1 0 µdenotes the class of all families of nonempty subsets b DH1 0={D(t)\H1 0(⌦) : t2R}, where b D={D(t):t2R}2D L2 µ. Now, from the existence of a pullback DL2 µ-absorbing family (cf. Proposition 2.13) and the regularising e↵ect of the equation (cf. Theorem 2.5), the following result is straightforward. Proposition 2.21. Under the assumptions of Lemma 2.19, the family b D0,H1 0={BL2(0,R 1/2 L2(t)) \H1 0(⌦) : t2R} belongs to DL2,H1 0 µand for any t2Rand any b D2D L2 µ,there exists ⌧3(b D,t)<tsuch that U(t, ⌧)D(⌧)⇢D0,H1 0(t)8⌧⌧3(b D,t). In particular, the family b D0,H1 0is pullback DL2,H1 0 µ-absorbing for the process U:R2 d⇥ H1 0(⌦)!H1 0(⌦). Proof. Let us fix t2Rand b D2D L2 µ.By Proposition 2.13, there exists ⌧0(b D,t)<t such that |U(t, ⌧)u⌧|2 2RL2(t)8u⌧2D(⌧)8⌧⌧0(b D,t). Moreover, thanks to the regularising e↵ect of the equation, when u⌧2L2(⌦), it holds that u(·;⌧,u⌧)2C((⌧,1); H1 0(⌦)).As a result, U(t, ⌧)u⌧2H1 0(⌦) if t>⌧. Therefore, it satisfies U(t, ⌧)D(⌧)⇢H1 0(⌦)\BL2(0,R 1/2 L2(t)) 8⌧⌧0(b D,t), where ⌧3(b D,t)=⌧0(b D,t). Finally, as a consequence of b D0belonging to DL2 µ,b D0,H1 02D L2,H1 0. µ Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 71 The following result establishes that the process Udefined on H1 0(⌦) as phasespace is pullback DL2,H1 0 µ-asymptotically compact. To that end, we apply again an energy method analogous to the one we used in Proposition 2.15. We reproduce it here just for the sake of completeness. Proposition 2.22. Under the assumptions of Lemma 2.19, the process U:R2 d⇥ H1 0(⌦)!H1 0(⌦)is pullback DL2,H1 0 µ-asymptotically compact. Proof. Let us fix t2R,a family b DH1 02D L2,H1 0 µ,a sequence {⌧n}⇢(1,t3] with ⌧n!1and u⌧n2DH1 0(⌧n)foralln.Wewillprovethatthesequence {u(t;⌧n,u ⌧n)}is relatively compact in H1 0(⌦).For short, we will denote un(·)= u(·;⌧n,u ⌧n). As a consequence of Lemma 2.19, it holds that there exists ⌧2(b D,t)<t3, such that ⌧n⌧2(b D,t)forallnn2,the sequence {un}nn2is bounded in L1(t2,t;H1 0(⌦)) \L2(t2,t;D()),and the sequences {a(l(un))un}nn2, {f(un)}nn2and {(un)0}nn2are bounded in L2(t2,t;L2(⌦)).Then, using the Aubin-Lions lemma, there exists u2L1(t2,t;H1 0(⌦))\L2(t2,t;D()) with u02L2(t2,t;L2(⌦)), such that for a subsequence (relabeled the same) it holds 8 > > > > > > > > > > > < > > > > > > > > > > > : un⇤ *uweakly-star in L1(t2,t;H1 0(⌦)), un*uweakly in L2(t2,t;D()), (un)0*u 0weakly in L2(t2,t;L2(⌦)), un!ustrongly in L2(t2,t;H1 0(⌦)), un(s)!u(s) strongly in H1 0(⌦) a.e. s2(t2,t), f(un)*f(u)weaklyinL2(t2,t;L2(⌦)), a(l(un))un*a(l(u))uweakly in L2(t2,t;L2(⌦)), (2.53) where the last two convergences have been obtained making use of [85, Lemme 1.3, p. 12]. Observe that u2C([t2,t]; H1 0(⌦)) and due to (2.53), usatisfies (2.6) in the interval (t2,t). Moreover, since {(un)0}nn2is bounded in L2(t2,t;L2(⌦)), it satisfies that {un}nn2is equicontinuous in L2(⌦) on [t2,t].Indeed, fixed ">0 and considering s1,s 22[t2,t]with|s2s1|<1, we have |un(s2)un(s1)|2 2=Zs2 s1 (un(⇠))0d⇠ 2 2 ✓Zs2 s1 |(un(⇠))0|2 2d⇠◆|s2s1| e⇢3(t)|s2s1|. Now, it just suffices to take "=min{"2/e⇢3(t),1}. From this and taking into account that {un}nn2is bounded in L1(t2,t;H1 0(⌦)) and the compactness of the embedding H1 0(⌦),!L2(⌦), applying the Arzela-Ascoli theorem we obtain un!ustrongly in C([t2,t]; L2(⌦)).(2.54) 72 2.4. Minimal attractors in H1-norm On the other hand, using that {un}nn2is bounded in C([t2,t]; H1 0(⌦)),we have that for any sequence {sn}⇢[t2,t]withsn!s⇤, un(sn)*u(s⇤)weaklyinH1 0(⌦),(2.55) where (2.54) has been used to identify the weak limit. If we prove un!ustrongly in C([t1,t]; H1 0(⌦)),(2.56) in particular, we have that the sequence {un(t)}is relatively compact in H1 0(⌦). To that end, we argue by contradiction. We suppose that there exist ">0,a sequence {tn}⇢[t1,t],without loss of generality converging to some t⇤,with kun(tn)u(t⇤)k2"8n1.(2.57) Now, applying (2.3), (2.4) and the Cauchy inequality to the energy equality (2.7), the estimate kz(s)k2 2kz(r)k2 2+2C2 f|⌦| m(sr)+2C2 f mZs r |z(⇠)|2 2d⇠+1 mZs r |h(⇠)|2 2d⇠ holds with zreplaced by uor any unfor all t2rst. Then, we define the following functions Jn(s)=kun(s)k2 22C2 f|⌦| ms2C2 f mZs t2 |un(r)|2 2dr 1 mZs t2 |h(r)|2 2dr, J(s)=ku(s)k2 22C2 f|⌦| ms2C2 f mZs t2 |u(r)|2 2dr 1 mZs t2 |h(r)|2 2dr. It is clear from the regularity of uand all unthat these functions are continuous on [t2,t]. In addition, using the above inequality it is not difficult to prove that J and all Jnare non-increasing functions on [t2,t].Moreover, from (2.53), it holds Jn(s)!J(s)a.e.s2(t2,t). Hence, there exists a sequence {˜ tk}⇢(t2,t ⇤)suchthat˜ tk!t⇤when k!1 and lim n!1 Jn(˜ tk)=J(˜ tk)8k1. Consider ✏>0 fixed. Since the function Jis continuous on [t2,t], there exists k(✏)1 such that |J(˜ tk)J(t⇤)|<✏ 28kk(✏). Now, we consider n(✏)1 such that tn˜ tk(✏)and |Jn(˜ tk(✏))J(˜ tk(✏))|<✏ 2nn(✏). Chapter 2. Non-autonomous nonlocal parabolic equation with sublinear terms 73 Then, since all the functions Jnare non-increasing, for all nn(✏) Jn(tn)J(t⇤)Jn(˜ tk(✏))J(t⇤) |Jn(˜ tk(✏))J(t⇤)| |Jn(˜ tk(✏))J(˜ tk(✏))|+|J(˜ tk(✏))J(t⇤)| <✏ 2+✏ 2=✏. Then, lim supn!1 Jn(tn)J(t⇤). Thus, it satisfies that lim supn!1 kun(tn)k2 ku(t⇤)k2which, together with (2.55) applied to the sequence {tn},allowustoprove that un(tn)convergetou(t⇤) strongly in H1 0(⌦), in contradiction with (2.57). Therefore, (2.56) holds. As a consequence of the above results, we obtain the existence of minimal pullback attractors for the process U:R2 d⇥H1 0(⌦)!H1 0(⌦). Theorem 2.23. Suppose that the function ais locally Lipschitz and (2.3) holds, f2C(R)fulfils (2.4), (2.5) and (2.30), l2L2(⌦)and h2L2 loc(R,L 2(⌦)) verifies (2.32) for some µ2(0,2(1m↵)). Then, there exist the minimal pullback DH1 0 F- attractor ADH1 0 F and the minimal pullback DL2,H1 0 µ-attractor ADL2,H1 0 µ for the process U:R2 d⇥H1 0(⌦)!H1 0(⌦).In addition, the following relationship holds ADH1 0 F (t)⇢A DL2 F(t)⇢A DL2 µ(t)=ADL2,H1 0 µ (t)8t2R,(2.58) In particular, we have the following pullback attraction result in H1 0(⌦), lim ⌧!1 distH1 0(U(t, ⌧)D(⌧),ADL2 µ(t)) = 0 8t2R8b D2D L2 µ.(2.59) Finally, if moreover hsatisfies sup s0✓eµs Zs 1 eµr|h(r)|2 2dr◆<1,(2.60) then the following chain of equalities holds ADH1 0 F(t)=ADL2 F(t)=ADL2 µ(t)=ADL2,H1 0 µ (t)8t2R, and for any B2D L2 F, lim ⌧!1 distH1 0(U(t, ⌧)B,ADL2 F(t)) = 0 8t2R.(2.61) Proof. The existence of ADH1 0 F and ADL2,H1 0 µ is a consequence of Corollary 1.15. Indeed, the process Uis strong-weak continuous (cf. Proposition 2.18), DH1 0 F⇢D L2,H1 0 µ holds, and the existence of an absorbing family (cf. Proposition 2.21) and the asymptotic compactness (cf. Proposition 2.22) hold. 80 3.1. Setting of the problem and existence results Observe that the sequence {Pnf(un)}is bounded in Lq(⌧,T;Hk(⌦)) since the sequence {f(un)}is bounded in Lq(⌧,T;Lq(⌦)). We will show the details for the sake of completeness. It satisfies hPnf(un(t)),viD(Ak/2),D(Ak/2)=Z⌦ f(un(t))Pnvdx |f(un(t))|q|Pnv|p C|f(un(t))|qkPnvkD(Ak/2) C|f(un(t))|qkvkD(Ak/2), where C>0 is the constant of the continuous embedding D(As/2),!Lp(⌦). Notice that D(Ak/2),!Hk(⌦),!Lp(⌦), since ⌦is a bounded open set of class Ck, with k2 such that kN(p2)/(2p) (cf. [100, Proposition 6.18, p. 183], [57, Theorem 5, p. 323]). Then, we have ZT ⌧ kPnf(un(t))kq D(Ak/2)dt CqZT ⌧ |f(un(t))|q qdt. (3.12) Therefore, the sequence {Pnf(un)}is bounded in Lq(⌧,T;D(Ak/2)).Bearing in mind that Lq(⌧,T;D(Ak/2)) ,!Lq(⌧,T;Hk(⌦)) (cf. [100, Proposition 6.19, p. 184]), we have that the sequence {Pnf(un)}is bounded in Lq(⌧,T;Hk(⌦)), In addition we have that the sequence {Pnh}is bounded in L2(⌧,T;H1(⌦)), since ZT ⌧ kPnh(t)k2 ⇤dt ZT ⌧ kh(t)k2 ⇤dt. (3.13) Then, bearing in mind (3.11), (3.12), (3.13) and the equality @un @t(t)a(l(un(t)))un(t)=Pnf(un(t)) + Pnh(t)inHk(⌦) a.e. t2(⌧,T), it holds that the sequence {u0 n}is bounded in Lq(⌧,T;Hk(⌦)). Therefore, from compactness arguments and the Aubin-Lions lemma, there exist a subsequence of {un}(relabeled the same) and u2L1(⌧,T;L2(⌦))\L2(⌧,T;H1 0(⌦))\Lp(⌧,T;Lp(⌦)) with u02L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)), such that 8 > > > > > > > > > > > < > > > > > > > > > > > : un ⇤ *uweakly-star in L1(⌧,T;L2(⌦)), un*uweakly in L2(⌧,T;H1 0(⌦)), un*uweakly in Lp(⌧,T;Lp(⌦)), u0 n*u 0weakly in Lq(⌧,T;Hk(⌦)), un!ustrongly in L2(⌧,T;L2(⌦)), a(l(un))un*a(l(u))uweakly in L2(⌧,T;H1 0(⌦)), f(un)*f(u)weaklyinLq(⌧,T;Lq(⌦)), Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 81 for all T>⌧. The limits of the sequences {f(un)}and {a(l(un))un}have been obtained by arguing analogously as done in the proof of Theorem 2.4, making use of [85, Lemme 1.3, p. 12]. Then, (3.6) follows taking limit when n!1in (3.8) and bearing in mind that [n2NVnis dense in H1 0(⌦)\Lp(⌦). Finally, to prove the existence of a weak solution to (3.1), we only need to check that u(⌧)=u⌧,whichmakescompletesensesinceu2C([⌧,T]; L2(⌦)). To do it we argue analogously as in the proof of Theorem 2.4. Consider '2H1(⌧,T)fixedwith '(T)=0and'(⌧)6=0.Now, we multiply by 'in (3.8), integrate between ⌧and T, and pass to the limit. Comparing this limiting equation with the expression obtained multiplying (3.6) by 'and integrating between ⌧and T, we conclude that u(⌧)=u⌧. Step 2. Uniqueness of weak solution. Let u1and u2be two weak solutions to (3.1) corresponding to the initial datum u1 ⌧,u 2 ⌧2L2(⌦). From the energy equality, we deduce 1 2 d dt|u1(t)u2(t)|2 2+a(l(u1(t)))ku1(t)u2(t)k2 2 =[a(l(u2(t)))a(l(u1(t)))]((u2(t),u 1(t)u2(t)))+(f(u1(t))f(u2(t)),u 1(t)u2(t)) a.e. t2[⌧,T]. Now, using that each solution to (3.1) belongs to C([⌧,T]; L2(⌦)), we have that ui(t)2Sfor all t2[⌧,T]andi=1,2, where Sis a bounded subset of L2(⌦).In addition, taking into account that l2L2(⌦), we have that {l(ui(t))}t2[⌧,T]⇢[R, R] for i=1,2, for some R>0.Therefore, using (3.2), (3.3), the locally Lipschitz continuity of the function aand the Cauchy inequality (cf. [57, Appendix B, p. 622]), we have d dt|u1(t)u2(t)|2 2(La(R))2|l|2 2ku2(t)k2 2+4m⌘ 2m|u1(t)u2(t)|2 2a.e. t2(⌧,T), where La(R) is the Lipschitz constant of the function ain [R, R]. Then, uniqueness follows. Observe that thanks to uniqueness of weak solution to (3.1), the whole sequence {un}converges to uweakly in L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) and weakly-star in L1(⌧,T;L2(⌦)). Similarly, the whole sequence {u0 n}converges to u0weakly in Lq(⌧,T;Hk(⌦)). Now, the existence and uniqueness of strong solutions to (3.1) as well as the regularising e↵ect of the equation will be proved. Recall that the strong solution uof a reaction-di↵usion equation belongs to L2(⌧,T;H2(⌦)\H1 0(⌦))\C([⌧,T]; H1 0(⌦)) for all T>⌧(cf. [100, 5]). To obtain this regularity, it is necessary to prove first that u02L2(⌧,T;L2(⌦)) and later, as a consequence it holds that f(u)2L2(⌧,T;L2(⌦)) and u2C([⌧,T]; H1 0(⌦)). However, in this more complex framework, due to the presence of the nonlocal term, we cannot prove directly the regularity of u0.The reason is that it does not seem to provide useful information to multiply by u0 nthe equation of the Galerkin approximations to obtain the boundedness of the sequence 82 3.1. Setting of the problem and existence results {u0 n}in L2(⌧,T;L2(⌦)). In this case, it seems to be necessary to analyse first the regularity of f(u), which belongs in general to Lq(⌧,T;Lq(⌦), and as a consequence we will be able to prove that u0belongs to Lq(⌧,T;Lq(⌦)).Observe that this regularity is not enough to prove the continuity of the solution in H1 0(⌦) with the strong topology, only with the weak one (cf. [108, Theorem 2.1, p. 544] or [111, Lemma 3.3, p. 74]). To prove the existence of a more regular solution and the regularising e↵ect of the equation, we need to establish a more regular setting. Instead of assuming that the function fsatisfies (3.3), we will suppose that f2C1(R)andverifies f0(s)⌘8s2R.(3.14) Observe that unlike what happens in Chapter 4, to prove this result we do not impose any restriction on dimension of the domain ⌦or additional strong assumptions on the function flike (4.13). Theorem 3.4. Assume that the function ais locally Lipschitz and fulfils (3.2), f2 C1(R)satisfies (3.4) and (3.14), h2L2 loc(R;L2(⌦)) and l2L2(⌦). Then, for any u⌧2L2(⌦), the weak solution u2L2(⌧+", T;H2(⌦)\H1 0(⌦)) \L1(⌧+", T;H1 0(⌦)) and u02Lq(⌧+", T;Lq(⌦)) for every ">0and T>⌧+". In addition, if the initial datum u⌧2H1 0(⌦), then there exists a unique strong solution uto (3.1) with u02Lq(⌧,T;Lq(⌦)). Proof. We split the proof into two steps. Step 1. Regularising e↵ect. Under the above assumptions, the existence of a unique weak solution to (3.1) is guaranteed by Theorem 3.3. Now, we will prove that u2L2(⌧+", T;H2(⌦)\H1 0(⌦)) \L1(⌧+", T;H1 0(⌦)) for all T>⌧+">⌧. Recall that from the energy equality for the Galerkin approximation un,atlight of (3.2), we obtained (3.9). Using (f(un(t)),u n(t)) |⌦|, (h(t),u n(t)) 1 21m|h(t)|2 2+m 2kun(t)k2 2, we deduce d dt|un(t)|2 2+mkun(t)k2 22|⌦|+1 1m|h(t)|2 2a.e. t2(⌧,T). Integrating between ⌧and T, we obtain |un(T)|2 2+mZT ⌧ kun(t)k2 2dt 2|⌦|(T⌧)+ 1 1mZT ⌧ |h(t)|2 2dt +|u⌧|2 2. In particular, ZT ⌧ kun(t)k2 2dt 2|⌦|(T⌧) m+1 1m2ZT ⌧ |h(t)|2 2dt +1 m|u⌧|2 2.(3.15) Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 83 Multiplying by j'nj in (3.8), summing from j=1ton, and using (3.2), we obtain 1 2 d dtkun(t)k2 2+m|un(t)|2 2(f(un(t)) f(0),un(t)) + (f(0) + h(t),un(t)) a.e. t2(⌧,T). Making use of the integration-by-parts formula, (3.14) and the Cauchy inequality, it holds d dtkun(t)k2 2+m|un(t)|2 22⌘kun(t)k2 2+2 m|h(t)|2 2+2|⌦|(f(0))2 ma.e. t2(⌧,T). (3.16) Integrating the previous expression between sand t,where⌧<stT, it holds kun(t)k2 2+mZt s |un(r)|2 2dr kun(s)k2 2+2⌘ZT ⌧ kun(r)k2 2dr +2 mZT ⌧ |h(r)|2 2dr +2|⌦|(f(0))2(T⌧) m.(3.17) In particular, kun(t)k2 2kun(s)k2 2+2⌘ZT ⌧ kun(r)k2 2dr +2 mZT ⌧ |h(r)|2 2dr +2|⌦|(f(0))2(T⌧) m. Integrating in sbetween ⌧and t, we obtain kun(t)k2 2(t⌧)(1 + 2⌘T2⌘⌧)ZT ⌧ kun(r)k2 2dr +2(T⌧) mZT ⌧ |h(r)|2 2dr +2|⌦|(f(0))2(T⌧)2 m. Then, taking into account (3.15), it holds that the sequence {un}is bounded in L1(⌧+", T;H1 0(⌦)) for all t2["+⌧,T]with"2(0,T ⌧). On the other hand, taking s=⌧+"and t=Tin (3.17), and using that {un}is bounded in L1(⌧+", T;H1 0(⌦)),we can deduce that {un}is bounded in L2(⌧+", T;H2(⌦)\H1 0(⌦)).Therefore, thanks to the uniqueness of weak solution, (un ⇤ *uweakly-star in L1(⌧+", T;H1 0(⌦)), un*uweakly in L2(⌧+", T;H2(⌦)\H1 0(⌦)). In addition, using (3.10) and the fact that {un}is bounded in L2(⌧+", T;H2(⌦)\ H1 0(⌦)), it satisfies that {a(l(un))un}is bounded in L2(⌧+", T;L2(⌦)).On the other hand, using (3.5) and taking into account that the sequence {un}is bounded in Lp(⌧,T;Lp(⌦)),it holds that {f(un)}is bounded in Lq(⌧,T;Lq(⌦)).Therefore, it verifies that u02Lq(⌧+", T;Lq(⌦)). Step 2. Strong solution. Assume that u⌧2H1 0(⌦). We need to check that u2L2(⌧,T;H2(⌦)\H1 0(⌦)) \L1(⌧,T;H1 0(⌦)) and u02Lq(⌧,T;Lq(⌦)) for all T>⌧. 84 3.1. Setting of the problem and existence results Integrating (3.16) between ⌧and t2[⌧,T], we obtain kun(t)k2 2+mZt s |un(r)|2 2dr ku⌧k2 2+2⌘ZT ⌧ kun(r)k2 2dr +2 mZT ⌧ |h(r)|2 2dr +2|⌦|(f(0))2(T⌧) m. Then, taking into account that {un}is bounded in L2(⌧,T;H1 0(⌦)) (see (3.15)), we deduce that {un}is bounded in L1(⌧,T;H1 0(⌦)) \L2(⌧,T;H2(⌦)\H1 0(⌦)). Thanks to the uniqueness of weak solutions, we have (un ⇤ *uweakly-star in L1(⌧,T;H1 0(⌦)), un*uweakly in L2(⌧,T;H2(⌦)\H1 0(⌦)). Thus, uis a strong solution in the sense of Definition 3.2. On the other hand, since @u @ta(l(u))u=f(u)+h(t)inLq(⌧,T;Lq(⌦)), it holds that u02Lq(⌧,T;Lq(⌦)). Observe that under the assumptions of Theorem 3.4, there exist functions f such that f(u)2L2(⌧,T;L2(⌦)) when uis the strong solution. Therefore, it can be proved that u2C([⌧,T]; H1 0(⌦)).For example, if we consider the functions f(s)=ss3or f(s)=|s|swith 3 when N= 3, it is not difficult to check that fverifies (3.4) and (3.14). In addition, using that u2L2(⌧,T;H2(⌦)\H1 0(⌦)) \ L1(⌧,T;H1 0(⌦)) and the fact that ⌦is an open set of class C2at least, it holds that u2L8(⌧,T;L8(⌦)) thanks to the Sobolev embeddings and the interpolation results [116, Lemma II.4.1, p. 72]. Hence, f(u)2L2(⌧,T;L2(⌦)) and then, it verifies u02L2(⌧,T;L2(⌦)). As a result u2C([⌧,T]; H1 0(⌦)) (see Section 3.4 below for more details). Now, we will show the Maximum Principle for (3.1), i.e. we will prove that if the initial datum u⌧0 a.e. ⌦then the solution of (3.1) fulfils that u(t)0for all t0 under suitable assumptions. This is a natural expected behaviour in a biological framework. Theorem 3.5. Assume that the function ais locally Lipschitz and fulfils (3.2), the function f2C(R;R+)satisfies (3.3) and (3.4), and the non-autonomous term h2L2(⌧,T;H1(⌦)) verifies hh(t),vi0a.e. t2(⌧,T)8v2H1 0(⌦)such that v0a.e. ⌦. Then, if the initial datum u⌧2L2(⌦)with u⌧0a.e. ⌦, the weak solution uto (3.1) fulfils that u(t)0for all t2[⌧,T]. Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 85 Proof. Under the above assumptions, thanks to Theorem 3.3, there exists a unique weak solution uto (3.1). Then we have @u @ta(l(u))u=f(u)+h(t)inL2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)). Observe that since uis the weak solution to (3.1), ubelongs to L2(⌧,T;H1 0(⌦))\ Lp(⌧,T;Lp(⌦)). Then, (u)+2L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)). Therefore, we have Zt ⌧ hdu ds(s),(u(s))+ids +Zt ⌧ a(l(u(s)))((u(s),(u(s))+))ds =Zt ⌧ (f(u(s)),(u(s))+)ds +Zt ⌧ hh(s),(u(s))+ids (3.18) for all t2[⌧,T]. Observe that hd dsu(s),(u(s))+i=1 2 d ds|(u(s))+|2 2a.e. s2(⌧,t), ((u(s),(u(s))+)) = (ru(s),r(u(s)+)=Z⌦ |r(u(s))+|2ds a.e. s2(⌧,t), (f(u(s)),(u(s))+)0 a.e. s2(⌧,t), hh(s),(u(s))+i0 a.e. s2(⌧,t). Therefore, taking this into account, from (3.18) we deduce 1 2|(u(t))+|2 2+Zt ⌧ a(l(u(s)))|r(u(s))+|2 2ds 1 2|(u(⌧))+|2 28t2[⌧,T]. Observe that Rt ⌧a(l(u(s)))|r(u(s))+|2 2ds 0and(u⌧)+= 0 since u⌧0 a.e. ⌦. Then, |(u(t))+|2 208t2[⌧,T]. Thus, (u(t))+= 0, i.e. u(t)0 for all t2[⌧,T]. 3.2 Analysis of the stationary problem In Chapter 2, using a corollary of the Brouwer fixed point theorem, we have shown the existence of stationary solutions to (3.1) when the function fis globally Lipschitz (cf. Theorem 2.7). However, when the function fis more general, it is not possible to argue in this way. In this section, the existence of at least one nontrivial stationary solution to (3.1) is analysed for the particular case f:⇥0,1 b⇤!Rgiven by f(s)=bs b3s3with b>0. Namely, we restrict ourselves to the case in which b=1forthesakeof simplicity. To do that, we make use of a result proved by Chipot & Corrˆea in [42] based on a fixed point argument. 86 3.2. Analysis of the stationary problem Thereupon, a conditional result between the solution to (3.1) and two (assumed to exist) stationary solutions to (3.1) is established in a new setting in the spirit of those appeared in [44, 45]. To do this, we argue similarly to [44, Lemma 4.1]. In what follows, since we are dealing with stationary solutions, we will assume that the function hdoes not depend on time, i.e. h2H1(⌦). Now we consider the elliptic problem ⇢a(l(u))u=f(u)+hin ⌦, u=0 on@⌦.(3.19) Definition 3.6. A solution to (3.19) is a function u⇤2H1 0(⌦)\Lp(⌦)which fulfils a(l(u⇤))((u⇤,v)) = (f(u⇤),v)+hh, vi8v2H1 0(⌦)\Lp(⌦). The analysis of the stationary solutions to (3.1) and their stability is a difficult problem due to the nonlinearity f(u). While many authors have been interested in (3.1) when fdoes not depend on the function u(for instance cf. [44, 45, 35, 48, 49, 50]), there are few studies in the more complex framework with f(u). Recently, Simsen & Ferreira have analysed a particular case of problem (3.1), which contains a unique stationary solution, the trivial one, and the exponential decay of the solutions of the evolution problem towards the stationary one has been established (cf. [106, Theorem 6]). Theorem 3.7 (Ferreira & Simsen).Assume that the function ais globally Lipschitz and satisfies (3.2), f(s)=g(s)|s|p2s, where gis globally Lipschitz (with Lipschitz constant ), g(0) = 0 and p2,h⌘0,l2L2(⌦)and m> 1 1.Then, for each initial datum u⌧2L2(⌦), the weak solution to (3.1) ufulfils |u(t)|2|u⌧|2e(m1)(t⌧)8t⌧. In particular, this means that in the above situation there exists a unique stationary solution, the trivial one. However, when h6⌘ 0orfsatisfies (3.4) with >0, it is worth noting that, with the same kind of estimates, the exponential decay of the solution of the evolution problem towards 0 does not necessarily hold. In this more general framework of problem (3.19), the uniqueness of solution is not guaranteed. Now, making use of a Chipot & Corrˆea’s result (cf. [42, Theorem 2.1]), the existence of at least one nontrivial solution to (3.19) will be proved. To that end, we need to assume that the function afulfils not only (3.2), but also 0<ma(s)M8s2R,(3.20) where Mis a positive constant. Theorem 3.8 (Chipot & Corrˆea).Suppose that the function a2C(R;R+)satisfies (3.20), 1M<1,f(s)=ss3,h⌘0and l2L2(⌦). Then, there exists at least one nontrivial solution to problem (3.19). Furthermore, any solution u⇤to (3.19) fulfils ku⇤k2↵:= ✓|⌦| 4m◆1/2 .(3.21) Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 87 Proof. Making use of the Schauder fixed point theorem, we will prove the existence of nontrivial stationary solutions. First, we need first to do some calculations. Analogously as it was denoted in the proof of Theorem 3.3, we represent by w1 the first normalized eigenfunction of , i.e. the solution to the problem 8 < : w1=1w1in ⌦, w12H1 0(⌦),w 1(x)0 a.e. x2⌦,Z⌦ (w1(x))2dx =1. Consider "0fixed such that 0 "0w1(x)1 a.e. x2⌦and 1Mf("0w1(x)) "0w1(x)a.e. x2⌦.(3.22) Observe that "0exists since w12L1(⌦) and 1M<1. In what follows we denote ˜u="0w1. From (3.20) and (3.22), we deduce that for all w2L2(⌦) ˜u="01w1f("0w1) M=f(˜u) Mf(˜u) a(l(w)) a.e. ⌦. Now we define the closed convex subset of L2(⌦) K={v2L2(⌦) : "0w1v1 a.e. ⌦}. Then, we are ready to define the map T:K!K w7! Tw =u⇤, where u⇤is the solution to the problem 8 < : u⇤+µu⇤ a(l(w)) =g(w) a(l(w)), u⇤2H1 0(⌦), (3.23) where g(s)=f(s)+µs and µis a positive constant such g0(s)0 for all s2(0,1). Observe that if Thas a fixed point, then it is a solution to (3.19). The application Tis well-defined, since u⇤2K. Namely, since gis a nondecreasing function in (0,1) and w2K, it holds (u⇤1) + µ a(l(w))(u⇤1) 0 a.e. ⌦, (˜uu⇤)+ µ a(l(w))(˜uu⇤)0 a.e. ⌦. In addition, since ˜u, u⇤2H1 0(⌦), we have u⇤10on@⌦, ˜uu⇤0on@⌦. 88 3.2. Analysis of the stationary problem Then, making use of the weak Maximum Principle, we obtain ˜uu⇤1 a.e. ⌦. Therefore, u⇤2K. To apply the Schauder fixed point Theorem, we need to check that Tis continuous and compact. Firstly, we will prove that the application Tis continuous. Let {wn}⇢Kbe such that wn!wstrongly in L2(⌦).(3.24) Observe that w2Ksince Kis a closed subset of L2(⌦).We want to prove that {u⇤ n},whereu⇤ n=Twnfor all n1, fulfils u⇤ n!u⇤strongly in L2(⌦). (3.25) Since u⇤and u⇤ nare solutions to (3.23) with wand wnrespectively, we have ((u⇤u⇤ n,v)) + µ a(l(w))(u⇤u⇤ n,v)=µ✓1 a(l(wn)) 1 a(l(w))◆(u⇤ n,v) +✓g(w) a(l(w)) g(wn) a(l(wn)),v◆, for all v2H1 0(⌦).Taking v=u⇤u⇤ nas a test function, and using (3.20), the fact that the sequence {u⇤ n}⇢Kand the Cauchy-Schwartz inequality, we obtain ku⇤u⇤ nk2 2+µ M|u⇤u⇤ n|2 2µ 1 a(l(wn)) 1 a(l(w))|u⇤u⇤ n|2 + g(w) a(l(w)) g(wn) a(l(wn))2 |u⇤u⇤ n|2|u⇤ n|2. Now, applying the Cauchy inequality, we have ku⇤u⇤ nk2 2Mµ 2 1 a(l(wn)) 1 a(l(w)) 2 |u⇤ n|2 2+M 2µ g(w) a(l(w)) g(wn) a(l(wn)) 2 2 . Observe that since l2L2(⌦) and a2C(R;R+), we deduce a(l(wn)) !a(l(w)).(3.26) Now, we will prove g(wn)!g(w) strongly in L2(⌦).(3.27) To that end, we only need to prove that f(wn)!f(w) strongly in L2(⌦).(3.28) Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 89 Since f2C1([0,1]) and (3.24) holds, we deduce f(wn)!f(w)a.e.⌦. Moreover, |f(wn(x))|2=|f(wn(x)) f(0)|2=|f0(⇠[0,1])|2|wn(x)|2a.e. ⌦. Therefore, making use of the Lebesgue Dominated Convergence theorem, we deduce (3.28). Then, from (3.26) and (3.27), (3.25) holds. Finally, we will prove that Tis compact. Observe that sup s2[0,1] |g(s)|sup s2[0,1] |f(s)|+µ=: Cg, where Cgis a positive constant, since fis continuous and µis a positive constant. Then, from this and (3.20), it satisfies ku⇤k2 21 a(l(w))(g(w),u ⇤) 1 mZ⌦ |g(w(x))||u⇤(x)|dx Cg|⌦|1/2 1/2 1mku⇤k2. Therefore, ku⇤k2Cg|⌦|1/2 1/2 1m. From this and taking into account that the injection H1 0(⌦),!L2(⌦) is compact, we deduce that the application Tis compact. Therefore, applying the Schauder fixed point theorem, we deduce that there exists at least one solution to (3.19). In addition, observe that any stationary solution satisfies a(l(u⇤))ku⇤k2 2=Z⌦⇥(u⇤(x))2(u⇤(x))4⇤dx |⌦| 4. Thus, using (3.20), (3.21) holds. Now, considering again the general form for the function fand under new suitable assumptions, we show that any stationary solution to (3.1) is positive provided that its existence is guaranteed. To that end, we suppose that the function a2C(R;R+) satisfies (3.2), f2C(R;R+), h2H1(⌦) fulfils h6⌘ 0,hh, vi08v2H1 0(⌦) such that v0 a.e. ⌦,(3.29) and l2L2(⌦). Then, we have the following Maximum Principle for (3.19). 96 3.3. Existence of pullback attractors in L2(⌦) Proof. Consider fixed t2Rand b D2D L2 µ. Thanks to Lemma 3.12 and (3.40), we have |U(t, ⌧)u⌧|2 2eµ(t⌧)|u⌧|2 2+2|⌦| µ+eµt 2mµ1 1Zt 1 eµskh(s)k2 ⇤ds (3.42) for all u⌧2D(⌧)and⌧t. Since b D2D L2 µ,there exists ⌧0(b D,t)<tsuch that eµ(t⌧)|u⌧|2 218u⌧2D(⌧)8⌧⌧0(b D,t).(3.43) Therefore, plugging (3.43) in (3.42), we obtain U(t, ⌧)D(⌧)⇢D0(t)8⌧⌧0(b D,t). Finally, using (3.41) it is not difficult to deduce that b D02D L2 µ. Then, to prove the existence of the minimal pullback attractor for the process U:R2 d⇥L2(⌦)!L2(⌦), we only need to check the pullback asymptotic compactness in L2(⌦) for the universe DL2 µ. To that end, we firstly establish the following result, which is the equivalent to Lemma 2.14 in the setting of this chapter. Observe that the proofs are very close. Nevertheless, we provide the details for the sake of completeness. Lemma 3.15. Under the assumptions of Proposition 3.14, for any t2Rand b D2 DL2 µ,there exists ⌧1(b D,t)<t2such that, for any ⌧⌧1(b D,t)and any u⌧2D(⌧), 8 > > > > > > > > < > > > > > > > > : |u(r;⌧,u⌧)|2 2⇢1(t)8r2[t2,t], Zr r1 ku(s;⌧,u⌧)k2 2ds ⇢2(t)8r2[t1,t], Zr r1 |u(s;⌧,u⌧)|p pds m 2↵2 ⇢2(t)8r2[t1,t], (3.44) where ⇢1(t)=1+2|⌦| µ+eµ(t2) 2mµ1 1Zt 1 eµskh(s)k2 ⇤ds, ⇢2(t)= 1 m✓⇢1(t)+2|⌦|+1 mmax r2[t1,t]Zr r1 kh(s)k2 ⇤ds◆. Proof. The first inequality in (3.44) as well as the expression of ⇢1follow by arguing as in the proof of Lemma 3.12, if ⌧⌧1(b D,t)<t2 (far enough pull back in time) due to our choice of tempered universe, taking into account (3.41). Notice that indeed this estimate also holds for the Galerkin approximations, which have already been used in Section 3.1. Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 97 For the other two inequalities in (3.44), we will prove them for the Galerkin approximations, and then, passing to the limit, we will obtain the same estimates for the solution. Multiplying by 'nj in (3.8), summing from j=1ton, and using (3.2), (3.4) and the Cauchy inequality, we deduce d ds|un(s)|2 2+mkun(s)k2 2+2↵2|un(s)|p p2|⌦|+1 mkh(s)k2 ⇤a.e. s>⌧. Now, integrating the above expression between r1andr,wehave |un(r)|2 2+mZr r1 kun(s)k2 2ds +2↵2Zr r1 |un(s)|p pds |un(r1)|2 2+2|⌦|+1 mZr r1 kh(s)k2 ⇤ds (3.45) for all ⌧r1. Then, from (3.45) we obtain for any n1 Zr r1 kun(s)k2 2ds ⇢2(t)8r2[t1,t]8u⌧2D(⌧)8⌧⌧1(b D,t),(3.46) where ⇢2(t) is given in the statement. Taking inferior limit in (3.46) and using the well-known fact that unconverge to u(·;⌧,u⌧)weaklyinL2(r1,r;H1 0(⌦)) for all r2[t1,t] (cf. Theorem 3.3), the second inequality in (3.44) holds. In addition, from (3.45) we also deduce that for any n1 Zr r1 |un(s)|p pds m 2↵2 ⇢2(t)8r2[t1,t]8u⌧2D(⌧)8⌧⌧1(b D,t). Now, taking inferior limit in the above expression and bearing in mind that un converge to u(·;⌧,u⌧)weaklyinLp(r1,r;Lp(⌦)) for all r2[t1,t](cf.Theorem 3.3), the last inequality in (3.44) holds. Now we will prove that the process Uis pullback DL2 µ-asymptotically compact using an energy method with continuous functions analogous to the one used in the proof of Proposition 2.15. Proposition 3.16. Under the assumptions of Proposition 3.14, the process U: R2 d⇥L2(⌦)!L2(⌦)is pullback DL2 µ-asymptotically compact. Proof. Let us fixed t2R,a family b D2D L2 µ,a sequence {⌧n}⇢(1,t2] with ⌧n!1,and u⌧n2D(⌧n)foralln.Ouraimistoprovethatthesequence {u(t;⌧n,u ⌧n)}is relatively compact in L2(⌦).For short we denote un(·)= u(·;⌧n,u ⌧n). Thanks to Lemma 3.15 we know that there exists ⌧1(b D,t)<t2 satisfying that, if n11 is such that ⌧n⌧1(b D,t)forallnn1,{un}nn1is bounded in L1(t2,t;L2(⌦)) \L2(t2,t;H1 0(⌦)) \Lp(t2,t;Lp(⌦)).Besides, from this, 98 3.3. Existence of pullback attractors in L2(⌦) making use of the continuity of the function aand bearing in mind that l2L2(⌦), {a(l(un))un}nn1is bounded in L2(t2,t;H1(⌦)). On the other hand, from (3.5), using that the sequence {un}is bounded in Lp(t2,t;Lp(⌦)),we deduce that {f(un)}nn1is bounded in Lq(t2,t;Lq(⌦)). As a consequence of the above uniform estimates, it holds that {(un)0}nn1is bounded in L2(t2,t;H1(⌦)) + Lq(t2,t;Lq(⌦)).Then, using the Aubin-Lions compactness Lemma, analogously as in the proof of Theorem 3.3, it holds that there exists u2L1(t2,t;L2(⌦)) \ L2(t2,t;H1 0(⌦)) \Lp(t2,t;Lp(⌦)),with u02L2(t2,t;H1(⌦)) + Lq(t 2,t;Lq(⌦)), such that for a subsequence (relabeled the same) it satisfies 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : un⇤ *uweakly-star in L1(t2,t;L2(⌦)), un*uweakly in L2(t2,t;H1 0(⌦)), un*uweakly in Lp(t2,t;Lp(⌦)), (un)0*u 0weakly in L2(t2,t;H1(⌦)) + Lq(t2,t;Lq(⌦)), un!ustrongly in L2(t2,t;L2(⌦)), un(s)!u(s) strongly in L2(⌦) a.e. s2(t2,t), f(un)*f(u)weaklyinLq(t2,t;Lq(⌦)), a(l(un))un*a(l(u))uweakly in L2(t2,t;H1(⌦)), (3.47) where the last two convergences have been obtained arguing in the same way as in the proof of Theorem 2.4. In addition, observe that u2C([t2,t]; L2(⌦)) and making use of (3.47), ufulfils (3.6) in the interval (t2,t). From (3.47) we can also deduce that {un}nn1is equicontinuous in H1(⌦) + Lq(⌦) on [t2,t]. To do this we argue similarly as in the proof of Proposition 2.15. Moreover, we have that the sequence {un}nn1is bounded in C([t2,t]; L2(⌦)) and the embedding L2(⌦),!H1(⌦) + Lq(⌦) is compact. Therefore, applying the Arzela-Ascoli theorem, we have (for another sequence, relabeled again the same) that un!ustrongly in C([t2,t]; H1(⌦) + Lq(⌦)).(3.48) Thanks to the boundedness of {un}nn1in C([t2,t]; L2(⌦)),for any sequence {sn}⇢[t2,t]withsn!s⇤, we obtain un(sn)*u(s⇤)weaklyinL2(⌦),(3.49) where we have used (3.48) to identify the weak limit. In this proof, we will show not only that the sequence {un(t)}nn1is relatively compact, but also the stronger convergence un!ustrongly in C([t1,t]; L2(⌦)).(3.50) We argue by contradiction. On the one hand, suppose that there exist ">0,a sequence {tn}⇢[t1,t], without loss of generality converging to some t⇤,with |un(tn)u(t⇤)|2"8n1.(3.51) Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 99 On the other hand, applying the Cauchy inequality, (3.2) and (3.4) to the energy equality (3.7), we deduce |z(s)|2 2|z(r)|2 2+2|⌦|(sr)+ 1 2mZs r kh(⇠)k2 ⇤d⇠8t2rst, where zmay be replaced by uor any un. Now we define the following functions Jn(s)=|un(s)|2 22|⌦|s1 2mZs t2 kh(r)k2 ⇤dr, J(s)=|u(s)|2 22|⌦|s1 2mZs t2 kh(r)k2 ⇤dr, which are continuous and non-increasing on [t2,t]. Moreover, observe that from (3.47) we obtain Jn(s)!J(s)a.e.s2(t2,t). Therefore, there exists a sequence {˜ tk}⇢(t2,t ⇤)suchthat˜ tk!t⇤when k!1 and such that the above convergence holds for any ˜ tk. Then consider an arbitrary value ✏>0 fixed. Since the function Jis continuous on [t2,t],there exists k(✏)1 such that |J(˜ tk)J(t⇤)|<✏ 28kk(✏). Now consider n(✏)1suchthat tn˜ tk(✏)and |Jn(˜ tk(✏))J(˜ tk(✏))|<✏ 28nn(✏). Therefore, as the functions Jnare non-increasing, making use of the previous estimates, we have Jn(tn)J(t⇤)Jn(˜ tk(✏))J(t⇤) |Jn(˜ tk(✏))J(t⇤)| |Jn(˜ tk(✏))J(˜ tk(✏))|+|J(˜ tk(✏))J(t⇤)| <✏ 2+✏ 2=✏8nn(✏). Since ✏>0 is arbitrary, it yields lim sup n!1 Jn(tn)J(t⇤).Therefore, taking into account the expressions of Jand all Jn,wededucethatlimsup n!1 |un(tn)|2 |u(t⇤)|2. Then, from this, together with (3.49), we conclude that un(tn)converge to u(t⇤) strongly in L2(⌦), which is contradictory with (3.51). Therefore, (3.50) holds. As a consequence, we have the main result of this section. The proof of this result is quite close to the proof of Theorem 2.16. We show the details for the sake of completeness. 100 3.4. Existence of pullback attractors in H1 0(⌦) Theorem 3.17. Assume that the function ais locally Lipschitz and (3.2) holds, f2C(R)satisfies (3.3) and (3.4), h2L2 loc(R;H1(⌦)) fulfils condition (3.41) for some µ2(0,21m)and l2L2(⌦). Then, there exist the minimal pullback DL2 F-attractor ADL2 Fand the minimal pullback DL2 µ-attractor ADL2 µfor the process U:R2 d⇥L2(⌦)!L2(⌦). Furthermore, the family ADL2 µbelongs to DL2 µand it holds ADL2 F(t)⇢A DL2 µ(t)⇢BL2(0,R 1/2 L2(t)) 8t2R. Besides, if the function hfulfils sup s0✓eµs Zs 1 eµrkh(r)k2 ⇤dr◆<1,(3.52) then ADL2 F(t)=ADL2 µ(t)8t2R. Proof. The existence of ADL2 µ,ADL2 Fand the first relation between both attractors is a consequence of Corollary 1.15. Indeed, the continuity of the process (cf. Proposition 3.11), the relationship DL2 F⇢D L2 µ, the existence of an absorbing family (cf. Proposition 3.14) and the asymptotic compactness (cf. Proposition 3.16) hold. The relation between the family ADL2 µand b D0is a direct consequence of Theorem 1.13. In addition, the family ADL2 µbelongs to DL2 µsince b D02D L2 µ,D0(t)isaclosed subset of L2(⌦) for all t2Rand the universe DL2 µis inclusion-closed. Finally, under assumption (3.52), we deduce that [tTRL2(t) is bounded for each T2R,whereRL2is given in Proposition 3.14. Thus, making use of Corollary 1.15, we deduce that both families of attractors coincide. 3.4 Existence of pullback attractors in H1 0(⌦) The goal of this section is to improve the results of the previous one, by establishing attraction in H1 0(⌦). In addition, we establish relationships between these new pullback attractors and those analysed in Theorem 3.17. Under the assumptions made in Section 3.1, namely in Theorem 3.4, we cannot guarantee the existence of a more regular energy equality for strong solutions (cf. Definition 3.2) because in general u0does not belong to L2(⌧,T;L2(⌦)). While in reaction-di↵usion equations, the regularity of u0can be obtained independently of the regularity of f(u) (cf. [5, Chapter 2, p. 32]), in nonlocal problems like (3.1), u0 inherits the regularity of f(u), which in general belongs to Lq(⌧,T;Lq(⌦)). To guarantee that f(u)2L2(⌧,T;L2(⌦)), it is enough for instance to assume that |f0(s)|Cfor all s2R, thanks to the mean value theorem, having uthe regularity of the weak solutions to (4.1). However, the sublinear case has already been studied with detail in Chapter 2. In this chapter, to study the asymptotic behaviour of the solutions making use of the theory of attractors, we assume f(u)2L2(⌧,T;L2(⌦)) 8u2L2(⌧,T;H2(⌦)\H1 0(⌦)) \L1(⌧,T;H1 0(⌦)).(3.53) Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 101 Although not every function satisfying (3.4) fulfils (3.53), there exists a wide range of functions which do. For instance, consider the function f(s)=ss3when N=3orf(s)=s|s|where 2(0,3] when N=3,2(0,2) when N=4, and 2(0,4/(N2)] when N5. Observe that using the Sobolev embeddings and interpolation results [116, Lemma II.4.1, p. 72], it can be checked that this last function also satisfies (3.53). In this case, we can deduce some information about the growth of f(u), since kf(u)k2 L2(⌧,T;L2(⌦)) Cfkuk2˜ b L1(⌧,T;H1 0(⌦))kuk2ˆ b L2(⌧,T;H2(⌦)\H1 0(⌦)),(3.54) for some ˆ b,˜ b,andCf>0. Namely, given ubelonging to L2(⌧,T;H2(⌦)\H1 0(⌦)) \ L1(⌧,T;H1 0(⌦)),using interpolation [116, Lemma II.4.1, p. 72] and the Sobolev embeddings, we deduce that ˆ b=(+1)✓and ˜ b=(+ 1)(1 ✓), where ✓2[0,1] is an interpolation exponent between Sobolev spaces, and kf(u)k2 L2(⌧,T;L2(⌦)) =kuk2+2 L2+2(⌧,T;L2+2(⌦)) kuk2˜ b L1(⌧,T;Lp(N,H1 0)(⌦))kuk2ˆ b L2(⌧,T;Lp(N,H2\H1 0)(⌦)) C2˜ b H1 0C2ˆ b H2\H1 0kuk2˜ b L1(⌧,T;H1 0(⌦))kuk2ˆ b L2(⌧,T;H2(⌦)\H1 0(⌦)) =Cfkuk2˜ b L1(⌧,T;H1 0(⌦))kuk2ˆ b L2(⌧,T;H2(⌦)\H1 0(⌦)), where CH1 0is the constant of the continuous embedding L1(⌧,T;H1 0(⌦)) ,! L1(⌧,T;Lp(N,H1 0)(⌦)) and CH2\H1 0is the constant of the embedding L2(⌧,T;H2(⌦)\ H1 0(⌦)) ,!L2(⌧,T;Lp(N,H2\H1 0)(⌦)). Then, assuming that the non-autonomous term h2L2(⌧,T;L2(⌦)), we can deduce that u02L2(⌧,T;L2(⌦)) for all T>⌧.Therefore, considering a more regular initial datum u⌧2H1 0(⌦), the associated strong solution u2C([⌧,1); H1 0(⌦)) and the following energy equality holds ku(t)k2 2+2Zt s a(l(u(r)))|u(r)|2 2dr =ku(s)k2 2+2Zt s (f(u(r)) + h(r),u(r))dr, (3.55) for all ⌧st. Observe that we can assume (3.54) thanks to the domain ⌦is smooth, since this allows to prove the existence of strong solutions and therefore, we can use this stronger regularity to make the most of the fact that f(u)2L2(⌧,T;L2(⌦)). Namely, we can make use of the Sobolev embedding L2(⌧,T;H2(⌦)\H1 0(⌦)) ,! L2(⌧,T;Lp(N,H2\H1 0)(⌦)) to deal with a wider range of functions f. Without assuming any smoothness conditions on the domain ⌦, as it is analysed in Chapter 4, we need to impose some restrictions on either the reaction term or the dimension of the domain ⌦and the reaction term, to prove the asymptotic behaviour of the solutions in H1 0(⌦).In addition, in Chapter 4, since we do not impose any smoothness conditions on the domain ⌦, we cannot use the cited Sobolev embedding (L2(⌧,T;H2(⌦)\H1 0(⌦)) ,!L2(⌧,T;Lp(N,H2\H1 0)(⌦))). 102 3.4. Existence of pullback attractors in H1 0(⌦) Observe that thanks to Theorem 3.4, the restriction of Uto R2 d⇥H1 0(⌦) defines a process into H1 0(⌦).Since no confusion arises, we will not modify the notation and continue denoting this process by U. Actually, this process defined on H1 0(⌦) as phase-space still fulfils properties to apply the results of Chapter 1. The following result shows that the process Uis strong-weak continuous in H1 0(⌦). Proposition 3.18. Suppose that the function ais locally Lipschitz and fulfils (3.2), f2C1(R)satisfies (3.4) and (3.14), h2L2 loc(R;L2(⌦)) and l2L2(⌦). Then, the process Uis strong-weak continuous in H1 0(⌦). Proof. Consider (t, ⌧)2R2 dfixed and let {uk ⌧}be a sequence of initial data which converges to u⌧strongly in H1 0(⌦). On the one hand, by Proposition 3.11, the map U(t, ⌧) is continuous from L2(⌦) into itself. Therefore, U(t, ⌧)uk ⌧!U(t, ⌧)u⌧strongly in L2(⌦).(3.56) On the other hand, observe that under the above assumptions we cannot guarantee that f(u)2L2(⌧,t;L2(⌦)), therefore we cannot use the stronger energy equality (3.55). Then, to solve this problem, we use the Galerkin approximations and pass to the limit by compactness arguments. Multiplying (3.8) by j'nj, summing from j=1ton, adding ±(f(0),un(t)) and making use of (3.2), we have 1 2 d dskun(s)k2 2+m|un(s)|2 2(f(un(s))f(0),un(s))+(f(0)+h(s),un(s)) a.e. s2(⌧,t). Integrating by parts and using (3.14) and the Cauchy inequality, we deduce d dskun(s)k2 22⌘kun(s)k2 2+1 2m|f(0) + h(s)|2 2a.e. s2(⌧,t). Then, integrating between ⌧and tand applying the Gronwall lemma, we have kun(t)k2 2✓kuk ⌧k2 2+1 2mZt ⌧ |f(0) + h(s)|2 2ds◆e2⌘(t⌧). Now, since the sequence {un}is bounded in L1(⌧,t;H1 0(⌦)), un(·;⌧,uk ⌧)*u(·;⌧,uk ⌧) weakly in L2(⌧,t;H1 0(⌦)) and u(·;⌧,uk ⌧)2C([⌧,t]; L2(⌦)) (cf. Theorem 3.3), taking into account [100, Lemma 11.2, p. 288], we have kU(t, ⌧)uk ⌧k2 2✓kuk ⌧k2 2+1 2mZt ⌧ |f(0) + h(s)|2 2ds◆e2⌘(t⌧). From this, together with (3.56), we obtain U(t, ⌧)uk ⌧*U(t, ⌧)u⌧weakly in H1 0(⌦). Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 103 The following result, which is analogous to Lemma 2.19, establishes some uniform estimates of the solutions in more regular norms in a finite-time interval up to time twhen the initial datum is shifted pullback far enough. To prove it, we will require not only assumption (3.53) but also a specific information of the growth of f(u), namely (3.54) as pointed out above. Furthermore, to simplify the statement, let us firstly introduce the following two quantities ⇢ext 1(t)=1+2|⌦| µ+eµ(t3) 2mµ1 1Zt 1 eµ⇠kh(⇠)k2 ⇤d⇠, ⇢ext 2(t)= 1 m✓⇢ext 1(t)+2|⌦|+1 mmax r2[t2,t]Zr r1 kh(⇠)k2 ⇤d⇠◆. (3.57) Lemma 3.19. Under the assumptions of Proposition 3.18, if falso fulfils (3.54) and hsatisfies (3.41) for some µ2(0,21m), then, for any t2Rand b D2D L2 µ, there exists ⌧2(b D,t)<t3such that for any ⌧⌧2(b D,t)and any u⌧2D(⌧),the following estimates hold 8 > > > > > > > > < > > > > > > > > : ku(r;⌧,u⌧)k2 2e⇢1(t)8r2[t2,t], Zr r1 |u(⇠;⌧,u⌧)|2 2d⇠e⇢2(t)8r2[t1,t], Zr r1 |u0(⇠;⌧,u⌧)|2 2d⇠e⇢3(t)8r2[t1,t], (3.58) with e⇢1(t)=(1+2⌘)⇢ext 2(t)+ 1 mmax r2[t2,t]Zr r1 |f(0) + h(⇠)|2 2d⇠, e⇢2(t)= 1 m✓e⇢1(t)+2⌘⇢ext 2(t)+ 1 mmax r2[t1,t]Zr r1 |f(0) + h(⇠)|2 2d⇠◆, e⇢3(t)=3(M(⇢ext 1(t),l))2e⇢2(t)+3Cf(e⇢1(t))˜ b(e⇢2(t))ˆ b+3 max r2[t1,t]Zr r1 |h(⇠)|2 2d⇠, where ˆ b,˜ b, Cfand M(⇢ext 1(t),l)are positive constants. Proof. Let us firstly observe that, analogously as we argued in Lemma 3.15, we may obtain uniform estimates for solutions in a longer time-interval (useful for our purposes). Namely, there exists ⌧2(b D,t)<t3 such that for any ⌧⌧2(b D,t)and any u⌧2D(⌧),we have |u(r;⌧,u⌧)|2⇢ext 1(t)8r2[t3,t],(3.59) Zr r1 ku(⇠;⌧,u⌧)k2 2d⇠⇢ext 2(t)8r2[t2,t], 104 3.4. Existence of pullback attractors in H1 0(⌦) where {⇢ext i}i=1,2are given in (3.57). Observe that these estimates also hold for the Galerkin approximations un(·;⌧,u⌧), which have already been used in Section 3.1. In addition, from the continuity of the function a,thefactthatl2L2(⌦) and the first inequality in (3.59), we deduce that there exits a positive constant M(⇢ext 1(t),l) such that a(l(un(r))) M(⇢ext 1(t),l)8r2[t3,t]8n1.(3.60) Similarly to the proof of Lemma 3.15, we will prove the inequalities in (3.58) for the Galerkin approximations and then, passing to the limit, for the solutions. Multiplying by j'nj in (3.8), summing from j=1tonand making use of (3.2), (3.14), and the Cauchy inequality, we deduce d d⇠kun(⇠)k2 2+m|un(⇠)|2 22⌘kun(⇠)k2 2+1 m|f(0) + h(⇠)|2 2a.e. ⇠>⌧.(3.61) Integrating between rand swith ⌧r1sr, we obtain in particular kun(r)k2 2kun(s)k2 2+2⌘Zr r1 kun(⇠)k2d⇠+1 mZr r1 |f(0) + h(⇠)|2 2d⇠. Integrating the last inequality w.r.t. sbetween r1andr,wehave kun(r)k2 2(1 + 2⌘)Zr r1 kun(s)k2 2ds +1 mZr r1 |f(0) + h(⇠)|2 2d⇠ for all ⌧r1. Therefore, from the estimate on the solutions by ⇢ext 2given above, one deduces that for any n1 kun(r;⌧,u⌧)k2 2e⇢1(t)8r2[t2,t]8u⌧2D(⌧)8⌧⌧2(b D,t),(3.62) where e⇢1(t) is given in the statement. Now, taking inferior limit in (3.62) and using the well-known fact that unconverge to u(·;⌧,u⌧)2C([t2,t]; H1 0(⌦)) weakly-star in L1(t2,t;H1 0(⌦)) (cf. Theorem 3.4), the first inequality in (3.58) holds. Now, integrating between r1andrin (3.61), we obtain in particular Zr r1 |un(⇠)|2 2d⇠ 1 m✓kun(r1)k2 2+2⌘Zr r1 kun(⇠)k2 2d⇠+1 mZr r1 |f(0) + h(⇠)|2 2d⇠◆ for all ⌧r1.Then, for any n1 Zr r1 |un(⇠)|2 2d⇠e⇢2(t)8r2[t1,t]8u⌧2D(⌧)8⌧⌧2(b D,t),(3.63) where e⇢2(t) is given in the statement. Then, taking inferior limit in (3.63) and bearing in mind the well-known fact that unconverge to u(·;⌧,u⌧)weaklyinL2(r 1,r;H2(⌦)\H1 0(⌦)) for all r2[t1,t] (cf. Theorem 3.4), the second inequality in (3.58) holds. Chapter 3. Non-autonomous nonlocal reaction-di↵usion equations 105 Now, taking into account that fsatisfies (3.54) and the previous estimates, we have Zr r1 |f(un(⇠))|2 2d⇠Cf(e⇢1(t))˜ b(e⇢2(t))ˆ b8r2[t1,t]8u⌧2D(⌧)8⌧⌧2(b D,t). (3.64) Finally, since for all ⌧r1 Zr r1 |u0 n(⇠)|2 2d⇠ 3Zr r1 |a(l(un(⇠)))un(⇠)|2 2d⇠+3Zr r1 |f(un(⇠))|2 2d⇠+3Zr r1 |h(⇠)|2 2d⇠, from (3.60), (3.63) and (3.64), we obtain for any n1 Zr r1 |u0 n(⇠)|2 2d⇠e⇢3(t)8r2[t1,t]8u⌧2D(⌧)8⌧⌧2(b D,t), where e⇢3is given in the statement. Then, taking inferior limit in the above expression and taking into account that u0 nconverge to u0(·;⌧,u⌧)weaklyinL2(r1,r;L2(⌦)) for all r2[t1,t], we obtain the last inequality in (3.58). Now, we introduce new universes which involve more regularity. Definition 3.20. For each µ>0,DL2,H1 0 µdenotes the class of all families of nonempty subsets b DH1 0={D(t)\H1 0(⌦) : t2R}, where b D={D(t):t2R}2D L2 µ. As a direct consequence of the regularising e↵ect of the equation when the function h2L2 loc(R;L2(⌦)) (cf. Theorem 3.4) and the existence of a family pullback DL2 µ-absorbing (cf. Proposition 3.14), the existence of an absorbing family in the universe DL2,H1 0 µalso holds. We omit the proof because it is identical to the proof of Proposition 2.21. Proposition 3.21. Suppose that the function ais locally Lipschitz and (3.2) holds, f2C1(R)fulfils (3.4) and (3.14), h2L2 loc(R;L2(⌦)) satisfies condition (3.41) for some µ2(0,21m)and l2L2(⌦). Then, the family b D0,H1 0={BL2(0,R 1/2 L2(t)) \ H1 0(⌦) : t2R}2D L2,H1 0 µand for any t2Rand b D2D L2 µ,there exists ⌧3(b D,t)<t such that U(t, ⌧)D(⌧)⇢D0,H1 0(t)8⌧⌧3(b D,t). In particular, the family b D0,H1 0is pullback DL2,H1 0 µ-absorbing for the process U:R2 d⇥ H1 0(⌦)!H1 0(⌦). Now, to prove the pullback asymptotic compactness of Uin H1 0(⌦) for the universe DL2,H1 0 µ, we apply an energy method similar to the one use to prove Proposition 2.22. Proposition 3.22. Under the assumptions of Lemma 3.19, the process U:R2 d⇥ H1 0(⌦)!H1 0(⌦)is pullback DL2,H1 0 µ-asymptotically compact. 112 4.1. Statement of the problem. Existence results A3) Bi(t, ·)isboundedinV0 i,i.e. kBi(t, v)k⇤ic(1 + kvkpi1 i)8v2Vi. A4) Bi(t, ·) is monotone, i.e. hBi(t, v)Bi(t, w),vwi+|vw|208v,w 2Vi. A5) Bi(t, ·)iscoercive,i.e. hBi(t, v),vi+|v|2+C(t)↵kvkpi i8v2Vi. Suppose given functions hi2Lp0 i(⌧,T;V0 i)fori=1,...,m, and an initial datum u⌧2H. In what follows we denote B(t, v)= m X i=1 Bi(t, v)8v2V, h(t)= m X i=1 hi(t). Now we consider the following problem 8 > > > > < > > > > : u2 m \ i=1 Lpi(⌧,T;Vi)8T>⌧, u0(t)+B(t, u(t)) = h(t)inD0(⌧,T;V0), u(⌧)=u⌧. (4.8) Then we have the following result (see [85, Th´eor`eme 1.4, p. 168]). Theorem 4.3. Under the above assumptions there exists a unique solution uto (4.8). In addition, this solution fulfils u2C([⌧,T]; H)and u02 m X i=1 Lp0 i(⌧,T;V0 i). Remark 4.4. To prove Theorem 4.3, it can be used any numerable family formed by linearly independent elements such that the vector space generated by this family is dense in V. 4.1.2 Existence and uniqueness of weak solutions In this section we are going to apply the method stated in Section 4.1.1 to a nonautonomous reaction-di↵usion equation whose di↵usion term is composed by the Chapter 4. A monotone iterative approach for nonlocal RD equations 113 Laplacian and a viscosity term which depends only on time. Namely, for each n1, we denote by unthe weak solution to (Pn)8 > > > > < > > > > : u2L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) \C([⌧,T]; L2(⌦)), d dt(u(t),v)+a(l(un1(t)))((u(t),v)) = (f(u(t)),v)+hh(t),vi, u(⌧)=u⌧, where u0⌘0andunis the solution to (Pn)ifn1. Observe that the equation in (Pn)mustbeunderstoodinthesenseofD0(⌧,1)forallv2H1 0(⌦)\Lp(⌦). Theorem 4.5. Suppose that the function a2C(R;R+)satisfies (4.2), f2C(R) fulfils (4.3) and (4.4), h2L2 loc(R;H1(⌦)) and l2L2(⌦).Then, for any u⌧2L2(⌦) there exists a unique weak solution unto (Pn) for all n1. Proof. The existence and uniqueness of solution to problem (Pn)isduetoTheorem 4.3. Namely, take H=L2(⌦), V1=H1 0(⌦), p1=2,V 2=Lp(⌦) and p2=p,and define B1(t, v)=a(l(un1(t)))v8v2H1 0(⌦)8t2(⌧,T), B2(t, v)=f(v)8v2Lp(⌦)8t2(⌧,T), h1(t)=h(t)8t2(⌧,T), h2(t)=0 8t2(⌧,T). Then, making use of the fact that a2C(R;R+) satisfies (4.2) and l2L2(⌦), we have that B1fulfils A1)-A5). Analogously, using f2C(R) fulfils (4.3) and (4.4), it is not difficult to check that B2satisfies A1)-A5). As a result, there exists a unique solution to (Pn)foralln1. Now we are ready to prove the existence of weak solutions to (4.1). Observe that this result has been proved without assuming any smoothness condition on the domain ⌦. Theorem 4.6. Suppose that the function ais locally Lipschitz and satisfies (4.2), f2C(R)fulfils (4.3) and (4.4), h2L2 loc(R;H1(⌦)) and l2L2(⌦).Then, for each u⌧2L2(⌦), there exists a unique weak solution to problem (4.1), which is denoted by u(·)=u(·;⌧,u⌧). Moreover, this solution behaves continuously in L2(⌦)w.r.t. initial data. Proof. We split the proof into two steps. Step 1. Existence of weak solution. Consider u0⌘0, and defining unthe solution to (Pn)(cf.Theorem4.5),wehave 1 2 d dt|un(t)|2 2+a(l(un1))kun(t)k2 2=(f(un(t)),u n(t)) + hh(t),u n(t)ia.e. t2(⌧,T). 114 4.1. Statement of the problem. Existence results Now, making use of (4.2) and (4.4), we obtain d dt|un(t)|2 2+mkun(t)k2 2+2↵2|un(t)|p p2|⌦|+1 mkh(t)k2 ⇤a.e. t2(⌧,T). Integrating between ⌧and t2[⌧,T], |un(t)|2 2+mZt ⌧ kun(s)k2 2ds+2↵2Zt ⌧ |un(s)|p pds |u⌧|2 2+2k|⌦|(T⌧)+ 1 mZT ⌧ kh(s)k2 ⇤ds. Therefore, the sequence {un}is bounded in L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦)) \ Lp(⌧,T;Lp(⌦)).From this, we deduce that there exists a positive constant C1such that |un(t)|2C18t2[⌧,T]8n1. Then, using that the function a2C(R;R+)andl2L2(⌦), there exists a positive constant MC1such that a(l(un1(t))) MC18t2[⌧,T]8n1. Therefore, {a(l(un1))un}is bounded in L2(⌧,T;H1(⌦)). Moreover, making use of the boundedness of {un}in Lp(⌧,T;Lp(⌦)) and (4.5), we obtain that the sequence {f(un)}is bounded in Lq(⌧,T;Lq(⌦)) (where pand qare conjugate exponents). Taking this into account, we deduce that the sequence {(un)0}is bounded in L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)). Therefore, using the Aubin-Lions lemma, there exist a subsequence of {un}(relabeled the same), a function u2L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦))\Lp(⌧,T;Lp(⌦)) with u02L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)) such that 8 > > > > > > > > > > > < > > > > > > > > > > > : un⇤ *uweakly-star in L1(⌧,T;L2(⌦)), un*uweakly in L2(⌧,T;H1 0(⌦)), un*uweakly in Lp(⌧,T;Lp(⌦)), un!ustrongly in L2(⌧,T;L2(⌦)), (un)0*u 0weakly in L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)), f(un)*f(u)weaklyinLq(⌧,T;Lq(⌦)), a(l(un1))un*a(l(u))uweakly in L2(⌧,T;H1(⌦)), (4.9) where the limits of the sequences {f(un)}and {a(l(un))un}have been obtained applying [85, Lemme 1.3, p. 12] (see Theorem 2.4 for more details). Thereupon, we will show that ufulfils (4.6) for all v2H1 0(⌦)\Lp(⌦).Consider fixed T>⌧and '2D(⌧,T).Since unis a solution to (Pn), it satisfies for all n1 ZT ⌧ (un(t),v)'0(t)dt+ZT ⌧ a(l(un1(t)))hun(t),vi'(t)dt=ZT ⌧ hf(un(t))+h(t),vi'(t)dt. Taking limit when n!1in the previous expression and making use of (4.9), (4.6) holds. Chapter 4. A monotone iterative approach for nonlocal RD equations 115 Finally, to prove the existence of a weak solution to (4.1), we only need to check that u(⌧)=u⌧.Observe that this equality makes complete sense since u2 C([⌧,T]; L2(⌦)) (cf. [56, Th´eor`eme 2, p. 575]). To do this, consider fixed v2 H1 0(⌦)\Lp(⌦) and '2H1(⌧,T),with '(T)=0and'(⌧)6=0.Since unis a weak solution to (Pn), it holds d dt(un(t),v)+a(l(un1(t)))((un(t),v)) = (f(un(t)),v)+hh(t),via.e. t2(⌧,T). Now, multiplying by 'in the previous expression and integrating between ⌧and T, we have (u⌧,v)'(⌧)+ZT ⌧ a(l(un1(t)))((un(t),v))'(t)dt =ZT ⌧ (f(un(t)),v)'(t)dt +ZT ⌧ hh(t),vi'(t)dt. Thereupon, taking limit when n!1and considering (4.9), we obtain (u⌧,v)'(⌧)+ZT ⌧ a(l(u(t)))((u(t),v))'(t)dt =ZT ⌧ (f(u(t)),v)'(t)dt +ZT ⌧ hh(t),vi'(t)dt. (4.10) Otherwise, we deduce from (4.6) (u(⌧),v)'(⌧)+ZT ⌧ a(l(u(t)))((u(t),v))'(t)dt =ZT ⌧ (f(u(t)),v)'(t)dt +ZT ⌧ hh(t),vi'(t)dt. Comparing this with (4.10), we deduce that (u(⌧),v)'(⌧)=(u⌧,v)'(⌧). Finally, since '(⌧)6=0andH1 0(⌦)\Lp(⌦) is dense in L2(⌦), the equality u(⌧)=u⌧holds. Step 2. Uniqueness of weak solution and continuity w.r.t. initial data. This has already been proved in Theorem 3.3, namely Step 2. Remark 4.7. (i) Thanks to the uniqueness of weak solution to (4.1), if fulfils that the whole sequence {un}converges to uweakly in L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) and weakly-star in L1(⌧,T;L2(⌦)). Analogously, the whole sequence {(un)0}converges to u0weakly in L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)). (ii) The previous result can be extended to more general di↵usion terms like a(l(u))Au, where A=PN i,j=1 @ @xi⇣bij(x)@ @xj⌘, with bij 2L1(⌦)for all i, j =1...N and PN i,j=1 bij(x)⇠i⇠j⇣|⇠|2, where ⇣>0. 116 4.1. Statement of the problem. Existence results 4.1.3 Strong solutions and regularising e↵ect In this section, the existence and uniqueness of strong solutions and the regularising e↵ect of the equation are proved without assuming any smoothness condition on the domain ⌦as in Chapter 3. To do this, it will not be enough to assume u⌧2H1 0(⌦) and h2L2 loc(R;L2(⌦)) like in Chapter 2. To prove the existence of strong solutions to (4.1), we do not take an arbitrary Hilbert basis of L2(⌦) dense in H1 0(⌦)\Lp(⌦), we use the eigenfunctions of inH1 0(⌦), denoted by {wi}, to make the most of the compactness arguments. To that end, since we are not assuming that ⌦is regular, we impose some requirements on the dimension Nof the domain ⌦. Namely, we assume that N2p p2.(4.11) Then, the continuous embedding H1 0(⌦),!Lp(⌦) holds. Therefore, the eigenfunctions of inH1 0(⌦) can be used as a basis to prove the existence of weak solutions to (4.1). Now we denote Vn= span[w1,...,w n]foralln1 and for each integer n1, we represent by un(t)=un(t;⌧,u⌧) the Galerkin approximation of the solution u(t, ⌧;u⌧)to(4.1),whichisgivenby un(t)= n X j=1 'nj(t)wj, and it is the local solution to 8 > < > : d dt(un(t),w j)+a(l(un(t)))((un(t),w j)) = (f(un(t)),w j)+(h(t),w j),t2(⌧,1), (un(⌧),w j)=(u⌧,w j),j=1,...,n. (4.12) Then we are ready to establish the existence of strong solutions and the regularising e↵ect of the equation. We omit the proof because it is identical to that of Theorem 3.4. Theorem 4.8. Suppose that ⌦is a bounded open subset of RNwith Nfulfilling (4.11), the function ais locally Lipschitz and satisfies (4.2), f2C1(R)fulfils (4.3) and (4.4), h2L2 loc(R;L2(⌦)) and l2L2(⌦). Then for each u⌧2L2(⌦),there exists a unique weak solution uto problem (4.1) which belongs to L2(⌧+", T;D()) \ C((⌧,T]; H1 0(⌦)) for every ">0and T>⌧+".In addition, if u⌧2H1 0(⌦), the existence of a strong solution uto (4.1) is guaranteed with u02Lq(⌧,T;Lq(⌦)). Another possible choice to prove the regularising e↵ect of the equation and the existence of strong solutions without supposing either requirements on the domain ⌦(cf. Chapter3)orrestrictionsonthedimensionofthedomain,likeintheprevious result, is to assume restrictive conditions on the function fwhich guarantee that f(u)2L2(⌧,T;L2(⌦)) for all u2L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦)).To that end, we assume that f2C(R) such that |f(s)|C(1 + |s|+1)8s2R,(4.13) Chapter 4. A monotone iterative approach for nonlocal RD equations 117 where =2/N if N3. The fact that f(u)2L2(⌧,T;L2(⌦)) is obtained applying interpolation results (cf. [116, Lemma II.4.1, p. 72]) to the Sobolev spaces L1(⌧,T;L2(⌦)) and L2(⌧,T;H1 0(⌦)). Namely, when u2L1(⌧,T;L2(⌦))\ L2(⌧,T;H1 0(⌦)), we obtain ZT ⌧Z⌦ |f(u(x, t))|2dxdt2ZT ⌧Z⌦C2+|u(x, t)|2+2dxdt (4.14) 2C2|⌦|(T⌧)+2C2kuk2ˆ b L1(⌧,T;L2(⌦))kuk2˜ b L2(⌧,T;Lp(N,H1 0)) 2C2h|⌦|(T⌧)+(CI(N))2˜ bkuk2ˆ b L1(⌧,T;L2(⌦))kuk2˜ b L2(⌧,T;H1 0(⌦))i, where ˆ b=(1✓)(+1),˜ b=✓(+1),C I(N) is the constant of the continuous embedding of H1 0(⌦) into Lp-spaces and ✓2[0,1]. Observe that it is not necessary to impose smoothness conditions on the domain ⌦to make use of the cited Sobolev embeddings (cf. [20, Remarque 21, p.173]). Remark 4.9. (i) When N=1,2,can be any positive value since the solution u2L1(⌧,T;H1 0(⌦)) ,!Ls(⌧,T;Ls(⌦)) with s=1when N=1and s<1when N=2. Therefore, the norm of uin L2+2(⌧,T;L2+2(⌦)) can be bounded making use of the more regular spaces L1(⌧,T;L1(⌦)or Ls(⌧,T;Ls(⌦)) with s<1. (ii) Observe that there exist functions which are not all sublinear that fulfil (4.13). For example, it can be considered f(s)=sp|s|when N=4. Theorem 4.10. Suppose that the function ais locally Lipschitz and satisfies (4.2), f2C(R)fulfils (4.3), (4.4) and (4.13), h2L2 loc(R;L2(⌦)) and l2L2(⌦). Then for each u⌧2L2(⌦),there exists a unique weak solution uto problem (4.1) which belongs to L2(⌧+", T;D()) \C((⌧,T]; H1 0(⌦)) for every ">0and T>⌧+". In addition, if u⌧2H1 0(⌦), the existence of a strong solution uto (4.1) holds. In addition, u2C([⌧,1); H1 0(⌦)) and it fulfils the energy equality ku(t)k2 2+2Zt s a(l(u(r)))|u(r)|2 2dr =ku(s)k2 2+2Zt s (f(u(r)) + h(r),u(r))dr (4.15) for all ⌧st. Proof. We split the proof into two steps. In the first one, we will show the regularising e↵ect of the equation. Finally, in the second step, we will prove the existence of strong solutions together with the strong energy equality (4.15). Step 1. Regularising e↵ect. Analogously to what it was done in Theorem 2.4, it proves that the sequence {un},whichisboundedinL2(⌧,T;H1 0(⌦)) \ L1(⌧,T;L2(⌦)) \Lp(⌧,T;Lp(⌦)), converges to u, the weak solution to (4.1). In this case, we have to take into account that thanks to (4.13), {f(un)}is bounded in L2(⌧,T;L2(⌦)) (see (4.14) above). Thereupon, we will prove that ubelongs to L2(⌧+", T;D()) \C((⌧,T]; H1 0(⌦)) for all T>⌧+">⌧. Multiplying by 'nj in (4.12) and summing from j=1ton,weobtain 1 2 d dt|un(t)|2 2+a(l(un(t)))kun(t)k2 2=(f(un(t)),u n(t)) + (h(t),u n(t)) a.e. t2(⌧,T). (4.16) 118 4.1. Statement of the problem. Existence results Applying (4.2) together with (f(un(t)),u n(t)) |⌦|, (h(t),u n(t)) 1 21m|h(t)|2 2+m 2kun(t)k2 2, to (4.16), it yields d dt|un(t)|2 2+mkun(t)k2 22|⌦|+1 1m|h(t)|2 2a.e. t2(⌧,T). Integrating between ⌧and T, in particular we obtain ZT ⌧ kun(t)k2 2dt 2|⌦|(T⌧) m+1 1m2ZT ⌧ |h(t)|2 2dt +1 m|u⌧|2 2.(4.17) Multiplying by j'nj in (4.12), summing from j=1ton, and making use of (4.2) and the Cauchy inequality (cf. [57, Appendix B, p. 622]), we obtain d dtkun(t)k2 2+m|un(t)|2 22 m|f(un(t))|2 2+2 m|h(t)|2 2a.e. t2(⌧,T).(4.18) Now integrating this expression between sand t,where⌧<stT, we have kun(t)k2 2+mZt s |un(r)|2 2dr kun(s)k2 2+2 mZT ⌧ |f(un(r))|2 2dr+2 mZT ⌧ |h(r)|2 2dr. (4.19) In particular, kun(t)k2 2kun(s)k2 2+2 mZT ⌧ |f(un(r))|2 2dr +2 mZT ⌧ |h(r)|2 2dr. Integrating the previous expression w.r.t. sbetween ⌧and t, bearing in mind that {f(un)}is bounded in L2(⌧,T;L2(⌦)) (cf. (4.14)) and (4.17), we obtain that {un} is bounded in L1(⌧+", T;H1 0(⌦)) with ">0.Taking s=⌧+"and t=Tin (4.19), we deduce that {un}is bounded in L2(⌧+", T;D()), thanks to the boundedness of {un}in L1(⌧+", T;H1 0(⌦)). As a consequence, it is not difficult to prove that {u0 n}is bounded in L2(⌧+", T;L2(⌦)).Therefore, thanks to the uniqueness of a weak solution, we deduce un*uweakly in L2(⌧+", T;D()), u0 n*u 0weakly in L2(⌧+", T;L2(⌦)). Then, since u2L2(⌧+", T;D()) and u02L2(⌧+", T;L2(⌦)) for any "2 (0,T ⌧),it satisfies that u2C((⌧,T]; H1 0(⌦)). Step 2. Strong solutions. Now, assuming that u⌧2H1 0(⌦), we will prove that the weak solution ubelongs to L2(⌧,T;D())\C([⌧,T]; H1 0(⌦)). Multiplying Chapter 4. A monotone iterative approach for nonlocal RD equations 119 by j'nj in (4.12), summing from j=1ton, using (4.2) and the Cauchy inequality, we deduce (4.18). Integrating between ⌧and t2[⌧,T],we have kun(t)k2 2+mZt ⌧ |un(s)|2 2ds ku⌧k2 2+2 mZT ⌧ |f(un(s))|2 2ds +2 mZT ⌧ |h(s)|2 2ds. Since {f(un)}is bounded in L2(⌧,T;L2(⌦)),the sequence {un}is bounded in L1(⌧,T;H1 0(⌦))\L2(⌧,T;D()).As a consequence, we have that {a(l(un))un} is bounded in L2(⌧,T;L2(⌦)). Now, we define the projection operator Pn:L2(⌦)! Vn 7! Pn j=1(, wj)wj. Since {wj:j1}is a special basis, Pnis non-expansive in L2(⌦). Therefore, {Pnh} and {Pnf(un)}are bounded in L2(⌧,T;L2(⌦)). As a consequence, the sequence {u0 n} is bounded in L2(⌧,T;L2(⌦)).Hence, as the weak solution is unique, it holds un*uweakly in L2(⌧,T;D()), u0 n*u 0weakly in L2(⌧,T;L2(⌦)). Then, since u2L2(⌧,T;D()) and u02L2(⌧,T;L2(⌦)),it satisfies that u2 C([⌧,T]; H1 0(⌦)) and (4.15) ([100, Theorem 7.2, p. 191]) . Corollary 4.11. The thesis of Theorem 4.10 also holds when the assumption (4.13) is weakened by taking =2/(N2) in (4.13) as long as f2C1(R)and the dimension of ⌦fulfils 3N2p/(p2). 4.2 Pullback attraction in L2-norm This section is devoted to studying the asymptotic behaviour of the solutions of (4.1). Namely, under the assumptions made on Section 4.1.2, we prove the existence of pullback attractors in the phase space L2(⌦).Although this result is not new in this PhD project, since the existence of these families has been proved in Theorem 3.17, the method applied to prove the pullback asymptotic compactness is, since until now we have used energy methods that make use of continuous and non-increasing functions (see Propositions 2.15, 2.22, 3.16 and 3.22 for more details). Now, in this chapter we apply the energy method used by Rosa in [101] with some variations due to the presence of the nonlocal operator in the di↵usion term. Thanks to Theorem 4.6, we can define a process U:R2 d⇥L2(⌦)!L2(⌦) as U(t, ⌧)u⌧=u(t;⌧,u⌧)8u⌧2L2(⌦)8⌧t, where u(·;⌧,u⌧) denotes the weak solution to (4.1). Now, we show that the process Uis continuous on L2(⌦). We omit the proof because it is straightforward, since the solution behaves continuously in L2(⌦) w.r.t. initial data (cf. Theorem 4.6). 120 4.2. Pullback attraction in L2-norm Proposition 4.12. Suppose that the function ais locally Lipschitz and fulfils (4.2), f2C(R)satisfies (4.3) and (4.4), h2L2 loc(R;H1(⌦)) and l2L2(⌦). Then, for any pair (t, ⌧)2R2 d, the map U(t, ⌧)is continuous from L2(⌦)into itself. Then we have the following result, which will be essential to build a suitable tempered universe in P(L2(⌦)). Observe that this result has already been proved in Lemma 3.12. Lemma 4.13. Under the assumptions of Proposition 4.12, if the initial datum u⌧ belongs to L2(⌦), the solution uto (4.1) fulfils for all µ2(0,2m1) |u(t)|2 2eµ(t⌧)|u⌧|2 2+2|⌦| µ+eµt 2mµ1 1Zt ⌧ eµskh(s)k2 ⇤ds 8t⌧.(4.20) Now, we define the following tempered universe in P(L2(⌦)). Definition 4.14. The class of all families of nonempty subsets b D={D(t):t2 R}⇢P(L2(⌦)) such that lim ⌧!1 eµ⌧sup v2D(⌧) |v|2 2!= 0 (4.21) is denoted by DL2 µfor all µ>0. Again, according to the notation in Chapter 1, we denote by DL2 Fthe universe of families (parameterized in time but constant for all t2R) of fixed nonempty bounded subsets of L2(⌦). Now, if hfulfils a suitable growth condition, the existence of an absorbing family is guaranteed. This result has already been proved in Proposition 3.14. Proposition 4.15. Under the assumptions of Proposition 4.12, if for some µ2 (0,2m1)the function h2L2 loc(R;H1(⌦)) also fulfils Z0 1 eµskh(s)k2 ⇤ds < 1,(4.22) the family b D0={D0(t):t2R}defined by D0(t)=BL2(0,R 1/2 L2(t)),where RL2(t)=1+2|⌦| µ+eµt 2mµ1 1Zt 1 eµskh(s)k2 ⇤ds, is pullback DL2 µ-absorbing for the process U:R2 d⇥L2(⌦)!L2(⌦)and belongs to DL2 µ. Then, to prove the existence of minimal pullback attractors in L2(⌦), we only need to check that the process Uis pullback DL2 µ-asymptotically compact. First, we prove the following continuity result. Chapter 4. A monotone iterative approach for nonlocal RD equations 121 Proposition 4.16. Under the assumptions of Proposition 4.12, if {un ⌧}⇢L2(⌦) satisfies that un ⌧*u ⌧weakly in L2(⌦),then for all T>⌧ 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : U(t, ⌧)un ⌧*U(t, ⌧)u⌧weakly in L2(⌦)8t2[⌧,T], U(·,⌧)un ⌧*U(·,⌧)u⌧weakly in L2(⌧,T;H1 0(⌦)), U(·,⌧)un ⌧*U(·,⌧)u⌧weakly in Lp(⌧,T;Lp(⌦)), U(·,⌧)un ⌧!U(·,⌧)u⌧strongly in L2(⌧,T;L2(⌦)), a(l(U(·,⌧)un ⌧))U(·,⌧)un ⌧*a(l(U(·,⌧)u⌧))U(·,⌧)u⌧weakly in L2(⌧,T;H1 0(⌦)), pa(l(U(·,⌧)un ⌧))U(·,⌧)un ⌧*pa(l(U(·,⌧)u⌧))U(·,⌧)u⌧weakly in L2(⌧,T;H1 0(⌦)), f(U(·,⌧)un ⌧)*f(U(·,⌧)u⌧)weakly in Lq(⌧,T;Lq(⌦)). (4.23) Proof. Consider (T,⌧)2R2 dfixed. For short, we denote by un(t)=U(t, ⌧)un ⌧and u(t)=U(t, ⌧)u⌧. Then, from the energy equality (4.7), applying (4.2), (4.4) and the Cauchy inequality, we obtain |un(t)|2 2+mZt ⌧ kun(r)k2 2dr+2↵2Zt ⌧ |un(r)|p pdr |un ⌧|2 2+2|⌦|(T⌧)+ 1 mZT ⌧ kh(r)k2 ⇤dr. Therefore, {un}is bounded in L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) \C([⌧,T]; L2(⌦)) for all T>⌧.From this we deduce that there exists a positive constant C1such that |un(t)|2C18t2[⌧,T]8n1. Then, using that the function a2C(R;R+)andl2L2(⌦), there exists a positive constant MC1such that a(l(un(t))) MC18t2[⌧,T]8n1.(4.24) Therefore, {pa(l(un))un}and {a(l(un))un}are bounded in L2(⌧,T;H1 0(⌦)). On the other hand, using (4.5) and taking into account the boundedness of {un}in Lp(⌧,T;Lp(⌦)), it satisfies that {f(un)}is bounded in Lq(⌧,T;Lq(⌦)).Besides, {un(T)}is bounded in L2(⌦).Therefore, as a consequence of the previous estimates and using the Aubin-Lions Lemma, there exist a subsequence of {un}(relabeled the same) and functions v2L1(⌧,T;L2(⌦))\L2(⌧,T;H1 0(⌦))\Lp(⌧,T;Lp(⌦)), 128 4.3. Pullback attraction in H1-norm for some µ2(0,2m1)and l2L2(⌦). Then, there exist the minimal pullback DL2 F-attractor ADL2 Fand the minimal pullback DL2 µ-attractor ADL2 µfor the process U:R2 d⇥L2(⌦)!L2(⌦). Besides, the family ADL2 µbelongs to DL2 µ,and the following relationships hold ADL2 F(t)⇢A DL2 µ(t)⇢BL2(0,R 1/2 L2(t)) 8t2R. In addition, if halso satisfies sup s0✓eµs Zs 1 eµrkh(r)k2 ⇤dr◆<1, then both attractors coincide, i.e. ADL2 F(t)=ADL2 µ(t)for all t2R. 4.3 Pullback attraction in H1-norm In this section, we will improve the results given in the previous section by proving attraction in H1 0(⌦) and establish relationships amongst these new pullback attractors and those given in Section 4.2 without assuming any condition of regularity on the domain ⌦as done in Chapter 3. To do this, we state the results in the setting of Theorem 4.10. Observe that although it is possible to prove the existence of strong solutions assuming that N2p/(p2) and without making any additional requirement on f(cf. Theorem 4.8), to study the asymptotic behaviour of the solution in H1-norm, we need to make use of stronger energy equalities and the continuity of the solution uin H1 0(⌦).To that end, we need that u02L2(⌧,T;L2(⌦)). Due to the nonlinearity created by the nonlocal operator in the di↵usion term, we cannot analyse the regularity of u0directly (multiplying the equation by u0), but u0inherits the regularity of a(l(u))u+f(u)+h.Therefore,f(u) must belong to L2(⌧,T;L2(⌦)). To that end, it is essential in this section to assume (4.13). Observe that the main result of this section, the existence of pullback attractors in H1 0(⌦), is not new in this PhD project, since the existence of these families has been proved in Theorem 3.23 under di↵erent assumptions. Nevertheless, the energy method applied here to prove the pullback asymptotic compactness is, since we make use of the flattening property. In the setting of Theorem 4.10, the restriction of Uto R2 d⇥H1 0(⌦) defines a process into H1 0(⌦). Since no confusion arises, we will not modify the notation and continue denoting this process by U. Making use of the results studied in Chapter 1, we prove the existence of pullback attractors in H1 0(⌦). First of all, the process Uis strong-weak in H1 0(⌦). Proposition 4.19. Assume that the function ais locally Lipschitz and (4.2) holds, f2C(R)fulfils (4.3), (4.4) and (4.13), h2L2 loc(R;L2(⌦)) and l2L2(⌦).Then, the process Uis strong-weak continuous in H1 0(⌦). Chapter 4. A monotone iterative approach for nonlocal RD equations 129 Proof. Consider (t, ⌧)2R2 dfixed. Let {un ⌧}⇢H1 0(⌦) be a sequence which converges to u⌧strongly in H1 0(⌦). By Proposition 4.12, the map U(t, ⌧) is continuous from L2(⌦) into itself, therefore U(t, ⌧)un ⌧!U(t, ⌧)u⌧strongly in L2(⌦). On the other hand, from (4.7) using (4.2), (4.4) and the Cauchy inequality, we obtain kU(·,⌧)un ⌧k2 L1(⌧,t;L2(⌦)) 2|⌦|(t⌧)+ 1 1mkhk2 L2(⌧,t;L2(⌦)) +|un ⌧|2 2, Zt ⌧ kU(s, ⌧)un ⌧k2 2ds 2|⌦|(t⌧) m+1 1m2khk2 L2(⌧,t;L2(⌦)) +1 m|un ⌧|2 2. Now, applying (4.2), the Cauchy inequality, (4.13) and interpolation results (see (4.14)) to the energy equality (4.15), we deduce kU(t, ⌧)un ⌧k2 2 kun ⌧k2 2+1 mkhk2 L2(⌧,t;L2(⌦)) +2C2|⌦|(T⌧) m +2C2(CI(N))2˜ b mkU(·,⌧)un ⌧k2ˆ b L1(⌧,t;L2(⌦))kU(·,⌧)un ⌧k2˜ b L2(⌧,t;H1 0(⌦)), where CI(N) is the constant of the continuous embedding of H1 0(⌦) into Lp-spaces, ˆ b=(1✓)(+1)and˜ b=✓(+1),with =2/N when N3and✓2[0,1]. Then, thanks to the previous estimates, {U(t, ⌧)un ⌧}is bounded in H1 0(⌦).Thus, by the uniqueness of the limit, we obtain U(t, ⌧)un ⌧*U(t, ⌧)u⌧weakly in H1 0(⌦). The following lemma is essential to prove the pullback flattening property. We establish uniform estimates of the solutions in a finite-time interval up to twhen the initial datum is shifted pullback far enough. The idea of the proof is similar to the proofs of Lemmas 3.15 and 3.19. We provide the details for the sake of completeness. Lemma 4.20. Under the assumptions of Proposition 4.19, if h2L2 loc(R;L2(⌦)) satisfies (4.22) for some µ2(0,21m), then for any t2Rand b D2D L2 µ,there exists ⌧1(b D,t)<t2such that for any ⌧⌧1(b D,t)and any u⌧2D(⌧) 8 > > > > > > < > > > > > > : |u(r;⌧,u⌧)|2 2⇢1(t)8r2[t2,t], Zr r1 ku(⇠;⌧,u⌧)k2 2d⇠⇢2(t)8r2[t1,t], ku(r;⌧,u⌧)k2 2⇢3(t)8r2[t1,t], (4.36) 130 4.3. Pullback attraction in H1-norm where ⇢1(t)=1+2|⌦| µ+eµ(t2) 2m1 1µZt 1 eµ⇠kh(⇠)k2 ⇤d⇠, ⇢2(t)=2|⌦| m+1 m⇢1(t)+ 1 m2max r2[t1,t]Zr r1 kh(⇠)k2 ⇤d⇠, ⇢3(t)=⇢2(t)+2C2|⌦| m+2C2(CI(N))2˜ b m(⇢1(t))ˆ b(⇢2(t))˜ b+1 mmax r2[t1,t]Zr r1 |h(⇠)|2 2d⇠, where ˆ b=(1✓)(+1),˜ b=✓(+1),C I(N)is the constant of the continuous embedding of H1 0(⌦)into Lp-spaces and ✓2[0,1]. Proof. From (4.21) we deduce that there exists ⌧1(b D,t)<t2 such that eµ(t2)eµ⌧|u⌧|2 218u⌧2D(⌧)8⌧⌧1(b D,t). The first inequality of (4.36) follows directly from (4.20) using the previous estimate. Making use of this estimate together with the energy equality (4.7), the second inequality follows arguing analogously as it was done in Lemma 3.15. Observe that these two estimates also holds for the Galerkin approximations. Finally, we will prove the third inequality of (4.36). Consider fixed ⌧⌧1(b D,t) and u⌧2D(⌧). From (4.15) for the Galerkin approximations, making use of (4.2) and the Cauchy inequality, we deduce kun(r;⌧,u⌧)k2 2kun(s;⌧,u⌧)k2 2+1 mZr r1 |f(un(⇠))|2 2ds +1 mZr r1 |h(⇠)|2 2ds with ⌧r1sr. Integrating the previous inequality w.r.t. son [r1,r] and taking into account (4.14), we obtain kun(r;⌧,u⌧)k2 2Zr r1 kun(s;⌧,u⌧)k2 2ds +1 mZr r1 |h(⇠)|2 2d⇠+2C2|⌦| m +2C2(CI(N))2˜ b mkun(·;⌧,u⌧)k2ˆ b L1(r1,r;L2(⌦))kun(·;⌧,u⌧)k2˜ b L2(r1,r;H1 0(⌦)) for all ⌧r1. Now, applying the two first inequalities of (4.36) to the previous expression we have kun(r;⌧,u⌧)k2 2⇢3(t)8r2[t1,t], where ⇢3(t) is given in the statement. Taking inferior limit in the above expression and using the well-known fact that unconverge to u(·;⌧,u⌧) weakly-star in L1(t 1,t;H1 0(⌦)) (cf. Theorem 4.10), the third inequality of (4.36) holds. Thereupon, we introduce the following universe in P(H1 0(⌦)). Chapter 4. A monotone iterative approach for nonlocal RD equations 131 Definition 4.21. For each µ>0,DL2,H1 0 µdenotes the class of all families of nonempty subsets b DH1 0={D(t)\H1 0(⌦) : t2R}, where b D={D(t):t2R}2D L2 µ. Observe that DH1 0 F⇢D L2,H1 0 µand DL2,H1 0 µis inclusion-closed. The existence of a pullback DL2,H1 0 µ-absorbing family follows directly from the regularising e↵ect of the equation (cf. Theorem 4.10) and the existence of a pullback DL2 µ-absorbing family (cf. Proposition 4.15). We omit the proof of this result because it is identical to that of Proposition 2.21. Proposition 4.22. Under the assumptions of Proposition 4.19, if h2L2 loc(R;L2(⌦)) also fulfils condition (4.22) for some µ2(0,21m), then, the family b D0,H1 0={BL2(0,R 1/2 L2(t)) \H1 0(⌦) : t2R} belongs to DL2,H1 0 µand for any t2Rand any b D2D L2 µ, there exists ⌧2(b D,t)<tsuch that U(t, ⌧)D(⌧)⇢D0,H1 0(t)8⌧⌧2(b D,t). In particular, the family b D0,H1 0is pullback DL2,H1 0 µ-absorbing for the process U:R2 d⇥ H1 0(⌦)!H1 0(⌦). Thereupon, we will prove that the process U:R2 d⇥H1 0(⌦)!H1 0(⌦) satisfies the pullback b D0,H1 0-flattening property. In fact, we will prove that Ufulfils the pullback b DH1 0-flattening property for any b DH1 02D L2,H1 0 µ. We will also use the following result, whose proof is analogous to that of [78, Lemma 12]. Lemma 4.23. If h2L2 loc(R;L2(⌦)) satisfies condition Z0 1 eµs|h(s)|2ds < 1(4.37) for some µ2(0,2m1), then for any t2R lim ⇢!1 e⇢tZt 1 e⇢s|h(s)|2 2ds =0. Then we have the following result (the idea of the proof is close to that in [65, Proposition 31]). Proposition 4.24. Under the assumptions of Proposition 4.19, if h2L2 loc(R;L2(⌦)) also fulfils (4.37) for some µ(0,21m), then, for any ">0and t2R, there exists n(", t)2Nsuch that for any b D2D L2 µ,the projection Pn:H1 0(⌦)!Vnsatisfies a) {PnU(t, ⌧)D(⌧):⌧⌧1(b D,t)}is bounded in H1 0(⌦), b) for all ⌧⌧1(b D,t)and u⌧2D(⌧), it fulfils that k(IPn)U(t, ⌧)u⌧k2<", 132 4.3. Pullback attraction in H1-norm where ⌧1(b D,t)is given in Lemma 4.20. In particular, the process Uon H1 0(⌦)satisfies the pullback b DH1 0-flattening property for any b DH1 02D L2,H1 0 µ. Proof. Let us fix ">0,t2Rand b D2D L2 µ. The first property given in the statement follows directly from the fact that Pn is non-expansive in H1 0(⌦), since {wj:j1}is a special basis (see Section 4.1.3) for more details) and the third inequality of (4.36). Thereupon, we will prove the second property. To that end consider fixed ⌧ ⌧1(b D,t),u ⌧2D(⌧), and define u(r):=U(r, ⌧)u⌧and qn(r):=u(r)Pnu(r).Then, using (4.2) and the Cauchy inequality, from (4.15) we deduce for each n1 d drkqn(r)k2 2+m|qn(r)|2 22 m|f(u(r))|2 2+2 m|h(r)|2 2a.e. r2(t1,t). Since |qn(r)|2 2n+1kqn(r)k2 2,from the above inequality we deduce d drkqn(r)k2 2+n+1mkqn(r)k2 22 m|f(u(r))|2 2+2 m|h(r)|2 2a.e. r2(t1,t).(4.38) Now multiplying by emn+1rin (4.38), integrating between t1andt, making use of (4.13) and (4.36), we have emn+1tkqn(t)k2 2emn+1(t1)kqn(t1)k2 2+2 mZt t1 emn+1r|h(r)|2 2dr +4C2emn+1t m2n+1 ⇥|⌦|+(C2 N⇢3(t))(+1)⇤ emn+1(t1)⇢3(t)+ 2 mZt 1 emn+1r|h(r)|2 2dr +4C2emn+1t m2n+1 ⇥|⌦|+(C2 N⇢3(t))(+1)⇤, where CNis the constant of the continuous embedding of H1 0(⌦) into L2+2(⌦), with =2/N when N3. Then, applying Lemma 4.23 and taking into account that n!1as n!1, there exists n=n(", t)2Nsuch that the second property holds. From the above result, the asymptotic compactness in the H1-norm yields (see Proposition 1.18 for more details). Proposition 4.25. Under the assumptions of Proposition 4.24, the process Uon H1 0(⌦)is pullback DL2,H1 0 µ-asymptotically compact. As a consequence of the previous results, we obtain the existence of minimal pullback attractors for the process Uon H1 0(⌦). Relationships amongst these new attractors and those given in Theorem 4.18 are also established. We omit the proof because it is similar to those of Theorems 2.23 and 3.23. Chapter 4. A monotone iterative approach for nonlocal RD equations 133 Theorem 4.26. Assume that the function ais locally Lipschitz and (4.2) holds, f2C(R)satisfies (4.3), (4.4) and (4.13), h2L2 loc(R;L2(⌦)) fulfils condition (4.37) for some µ2(0,21m)and l2L2(⌦). Then, there exist the minimal pullback DH1 0 F- attractor ADH1 0 F and the minimal pullback DL2,H1 0 µ-attractor ADL2,H1 0 µ for the process U:R2 d⇥H1 0(⌦)!H1 0(⌦). In addition, the following relationships holds ADH1 0 F (t)⇢A DL2 F(t)⇢A DL2 µ(t)=ADL2,H1 0 µ (t)8t2R, where ADL2 Fand ADL2 µare respectively the minimal pullback DL2 F-attractor and the minimal pullback DL2 µ-attractor for the process U:R2 d⇥L2(⌦)!L2(⌦), whose existence is guaranteed by Theorem 4.18. In particular, the following pullback attraction result holds lim ⌧!1 distH1 0(U(t, ⌧)D(⌧),ADL2 µ(t)) = 0 8t2R8b D2D L2 µ. Finally, if halso satisfies sup s0✓eµs Zs 1 eµr|h(r)|2 2dr◆<1, then ADH1 0 F (t)=ADL2 F(t)=ADL2 µ(t)=ADL2,H1 0 µ (t)8t2R. Furthermore, in this case, for any B2D L2 F lim ⌧!1 distH1 0(U(t, ⌧)B,ADL2 F(t)) = 0 8t2R. As a consequence of the previous result and as a complement of Corollary 4.11, we have the following result. Corollary 4.27. Assume that the function ais locally Lipschitz and (4.2) holds, f2C1(R)satisfies (4.3), (4.4) and (4.13) with =2/(N2) if 3N2p/(p2), h2L2 loc(R;L2(⌦)) fulfils condition (4.37) for some µ2(0,21m)and l2L2(⌦). Then, the thesis of Theorem 4.26 holds. Remark 4.28. Observe that if N=1,2, Corollary 4.27 holds without assuming any restriction on the positive value (see Remark 4.9 (i) for more details). Chapter 5 Abstract results on the theory of multi-valued processes and pullback attractors In the last few decades, many authors have been interested in analysing problems without uniqueness of solution because it allows to weaken the assumptions on the nonlinear functions which appears in the equation. In addition, for a wide range of problems, such as di↵erential inclusions, variational inequalities, control infinite-dimensional systems and some partial di↵erential equations such as the three-dimensional Navier-Stokes equations, the uniqueness of solution is not guaranteed. It is interesting to analyse what the asymptotic behaviour of the solutions of this type of problems is and to do it, many authors make use of the theory of multi-valued dynamical systems. For instance, in [96] Melnik & Valero study the existence of the compact global attractor for di↵erential inclusions in Hilbert spaces. In [6], Anguiano et al. analyse the existence of pullback attractors for non-autonomous reaction-di↵usion equations in some unbounded domains. For problems where terms with delay appear there exist also papers in this multi-valued framework. For example, in [89] Mar´ın-Rubio studies the existence of attractors corresponding to a general class of parameterized delay di↵erential equations posed in potentially di↵erent state spaces. In this chapter, we briefly recall some abstract results on multi-valued nonautonomous dynamical systems. Concepts such as multi-valued process, pullback absorbing family for a universe Dor pullback asymptotic compactness, amongst others, are recalled. In addition, some properties of the omega-limit set are analysed in this new framework together with a result which guarantees the existence of minimal pullback attractors. Besides, relationships between these families are established. All the results of this chapter can be found in [96, 89, 27, 94, 6, 5]. 135 136 5.1. Basic concepts 5.1 Basic concepts Let (X,dX)beametricspace. Definition 5.1. A multi-valued process (also called multi-valued non-autonomous dynamical system) Uon Xis a family of mappings U(t, ⌧):X7! P(X)for any pair (t, ⌧)2R2 d, such that (i) U(⌧,⌧)x={x}8⌧2R8x2X. (ii) U(t, ⌧)x⇢U(t, s)(U(s, ⌧)x)8⌧st8x2X, where U(t, ⌧)W:= [ y2W U(t, ⌧)y8W⇢X. Observe that if the relationship given in (ii) is an equality instead of an inclusion, the multi-valued process Uis called strict. Definition 5.2. A multi-valued process Uon Xis upper semicontinuous if the mapping U(t, ⌧)is upper semicontinuous from Xinto P(X)for all (t, ⌧)2R2 d,i.e. for any x2Xand for every neighborhood Nin Xof the set U(t, ⌧)x, there exists a value ">0such that U(t, ⌧)y⇢Nprovided that dX(x, y)<". Consider a family of nonempty sets b D0={D0(t):t2R}⇢P(X). Definition 5.3. A multi-valued process Uon Xis pullback b D0-asymptotically compact if for any t2Rand any sequences {⌧n}⇢(1,t]and {xn}⇢Xsuch that ⌧n!1and xn2D0(⌧n)for all n, it fulfils that any sequence {yn}is relatively compact in X, where yn2U(t, ⌧n)xnfor all n. Again, analogously as in Chapter 1, we denote the omega-limit set of the family b D0by ⇤(b D0,t)=\ st[ ⌧s U(t, ⌧)D0(⌧) X 8t2R. Lemma 5.4 (Sequential characterisation of the omega-limit set).It holds that y2⇤(b D0,t)if and only if there exist sequences {⌧n}⇢(1,t]and {yn}⇢X such that ⌧n!1,yn2U(t, ⌧n)D0(⌧n)for all n2N,and limn!1 yn=y. The following lemma will be very helpful to prove the existence of a pullback D-attractor for a multi-valued process Uin Section 5.2. This result was proved in [5, Lemma 3.9] and it is a generalization of [29, Theorem 6, Lemma 8]. We will show the proof for the sake of completeness. Lemma 5.5. If the multi-valued process Uis pullback b D0-asymptotically compact, then the omega-limit set ⇤(b D0,t)is nonempty, compact and lim ⌧!1 distX(U(t, ⌧)D0(⌧),⇤(b D0,t)) = 0 8t2R.(5.1) Chapter 5. Theory of multi-valued processes and pullback attractors 137 In addition, the family {⇤(b D0,t):t2R}is minimal in the sense that if there exists a family of closed sets b C={C(t):t2R}which fulfils lim ⌧!1 distX(U(t, ⌧)D0(⌧),C(t)) = 0, then ⇤(b D0,t)⇢C(t)for all t2R.Moreover, if the multi-valued process Uis upper semicontinuous with closed values, it holds ⇤(b D0,t)⇢U(t, ⌧)⇤(b D0,⌧)8(t, ⌧)2R2 d.(5.2) Proof. Consider t2Rfixed. First of all, we will check that the set ⇤(b D0,t) is nonempty. This is straightforward. Let {yn}be a sequence such that yn2U(t, ⌧n)D0(⌧n)foralln2N and {⌧n}⇢(1,t]convergingto1.As the multi-valued process Uis pullback b D0-asymptotically compact, there exists a convergent subsequence of {yn},i.e. ynk!y2⇤(b D0,t). Therefore, ⇤(b D0,t)isnonempty. Now, we will show that the set ⇤(b D0,t) is compact. Since it is closed, we only need to check that it is relatively compact. To do this, consider given a sequence {yn}⇢⇤(b D0,t).Making use of Lemma 5.4, for each yn2⇤(b D0,t), there exist ⌧n!1and zn2U(t, ⌧n)D0(⌧n), such that dX(yn,z n)<1 n.(5.3) Then, as the process Uis pullback b D0-asymptotically compact, there exists a convergent subsequence of {zn}. Therefore, making use of (5.3), the sequence {yn}is relatively compact. The next step consists in proving (5.1). To do this, we argue by contradiction. Assume that there exist ">0 and a sequence {yn}with yn2U(t, ⌧n)D0(⌧n)forall n2Nand {⌧n}⇢(1,t]convergingto1,suchthat dX(yn,⇤(b D0,t)) >"8n2N.(5.4) Now, since the multi-valued process Uis pullback b D0-asymptotically compact, there exists a subsequence of {yn}(relabeled the same) such that yn!y2⇤(b D0,t)(see Lemma 5.4), which is a contradiction with (5.4). In addition, the family {⇤(b D0,t):t2R}is minimal. Assume that b C={C(t): t2R}is a family of closed sets which fulfils lim ⌧!1 distX(U(t, ⌧)D0(⌧),C(t)) = 0.(5.5) Consider given y2⇤(b D0,t), we will show that y2C(t). On the one hand, from Lemma 5.4, we deduce that there exist sequences {⌧n}⇢(1,t]and{yn}⇢X with ⌧n!1and yn2U(t, ⌧n)D0(⌧n)foralln2N,suchthatyn!y. On the other hand, from (5.5) we deduce lim n!1 distX(U(t, ⌧n)D0(⌧n),C(t)) = 0. 144 6.1. Setting of the problem and existence result 6.1 Setting of the problem and existence result Let ⌦⇢RNbe a nonempty bounded open set and ⌧2R.Then,weconsiderthe following perturbed non-autonomous nonlocal reaction-di↵usion problem (P")8 > < > : du dt (1 ")a(l(u))u=f(u)+"h(t)in⌦⇥(⌧,1), u=0 on@⌦⇥(⌧,1), u(x, ⌧)=u⌧(x)in⌦, where "2[0,1), the function a2C(R;R+)andthereexistsapositiveconstantm such that 0<ma(s)8s2R,(6.1) and l2L(L2(⌦),R).The function f2C(R) and fulfils ↵1|s|pf(s)s↵2|s|p8s2R,(6.2) where ↵1,↵2and are positive constants and p2. Observe that from (6.2), it follows that there exists a constant >0 such that |f(s)|(|s|p1+1) 8s2R.(6.3) Analogously to Chapter 4, u⌧2L2(⌦) and h2L2 loc(R;H1(⌦)). From now on, we identify L2(⌦) with its dual. Therefore, the chain of compact and dense embedding H1 0(⌦)⇢L2(⌦)⇢H1(⌦) holds. Observe that as a result of the previous identification, l(u)isunderstoodas(l,u), but for short it is denoted by l(u). Now we are going to analyse the existence of weak solutions to (P"). Definition 6.1. A weak solution to the problem (P")is a function uthat belongs to L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)) for all T>⌧, with u(⌧)=u⌧, and such that for all v2H1 0(⌦)\Lp(⌦) d dt(u(t),v)+(1")a(l(u(t)))((u(t),v)) = (f(u(t)),v)+"hh(t),vi,(6.4) where the previous equation must be understood in the sense of D0(⌧,1). When uis a weak solution to (P"), making use of the continuity of a,l2L2(⌦), (6.3) and (6.4), we deduce that u02L2(⌧,T;H1(⌦))+Lq(⌧,T;Lq(⌦)) for any T>⌧ (where pand qare conjugate exponents). Therefore, u2C([⌧,1); L2(⌦)) and the initial datum in (P") makes sense. Furthermore, the following energy equality holds |u(t)|2 2+2(1")Zt s a(l(u(r)))ku(r)k2 2dr (6.5) =|u(s)|2 2+2Zt s (f(u(r)),u(r))dr +2"Zt s hh(r),u(r)idr for all ⌧st. Now, the existence of weak solution to (P") is proved. It is worth noting that no assumption of smoothness on ⌦is imposed. Chapter 6. Nonlocal reaction-di↵usion equations without uniqueness 145 Theorem 6.2. Assume that the function a2C(R;R+)fulfils (6.1), "2[0,1), f2C(R)satisfies (6.2), h2L2 loc(R;H1(⌦)) and l2L2(⌦). Then, for any ⌧2R and any u⌧2L2(⌦), there exists at least one weak solution to (P"). Proof. To prove this result we use the Galerkin approximations. Let {wj:j 1}⇢H1 0(⌦)\Lp(⌦) be a Hilbert basis of L2(⌦) such that Sn2NVn,whereVn:= span{w1,...,w n},isdenseinH1 0(⌦)\Lp(⌦).Now, consider T>⌧fixed. For each integer n1, we denote by un(t;⌧,u⌧)=Pn j=1 'nj(t)wj(for short denoted by un(t)) a local solution to 8 > < > : d dt(un(t),w j)+(1")a(l(un(t)))((un(t),w j))=hf(un(t))+"h(t),w jit2(⌧,1), (un(⌧),w j)=(u⌧,w j),j=1,...,n, (6.6) Multiplying (6.6) by 'nj(t), summing from j=1tonand making use of (6.1), we obtain 1 2 d dt|un(t)|2 2+(1")mkun(t)k2 2(f(un(t)),u n(t)) + "hh(t),u n(t)ia.e. t2(⌧,tn), (6.7) where (⌧,tn) is an interval of existence of solutions to (6.6) by the Carath´eodory Theorem (cf. [52, Theorem 1.1, p. 43]). From (6.2), (f(un(t)),u n(t)) |⌦|↵2|un(t)|p p. On the other hand, using the Cauchy inequality and taking into account that "2[0,1), we have hh(t),u n(t)i 1 2(1 ")mkh(t)k2 ⇤+(1 ")m 2kun(t)k2 2. Therefore, applying these two inequalities to (6.7) we obtain d dt|un(t)|2 2+(1")mkun(t)k2 2+2↵2|un(t)|p p2|⌦|+" (1 ")mkh(t)k2 ⇤ a.e. t2(⌧,tn). Now, integrating between ⌧and t2(⌧,tn), we deduce |un(t)|2 2+(1")mZt ⌧ kun(s)k2 2ds +2↵2Zt ⌧ |un(s)|p pds |u⌧|2 2+2⌦(T⌧)+ " (1 ")mZT ⌧ kh(s)k2 ⇤ds. From the above a priori estimate, we deduce that solutions to (6.6) are defined in the whole interval [⌧,T] and the sequence {un}is bounded in L1(⌧,T;L2(⌦)) \ L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)).From this, bearing in mind that for all n2N each un2C[⌧,T]; L2(⌦), it holds that there exists a positive constant C1>0such that |un(t)|2C18t2[⌧,T]8n1. 146 6.1. Setting of the problem and existence result Now making use of the continuity of the function aand l2L2(⌦), we deduce that there exists a positive constant MC1>0suchthat a(l(un(t))) MC18t2[⌧,T]8n1.(6.8) Taking this into account together with the boundedness of {un}in L2(⌧,T;H1 0(⌦)), we obtain that the sequence {a(l(un))un}is bounded in L2(⌧,T;H1(⌦)). On the other hand, using (6.3) and the boundedness of {un}in Lp(⌧,T;Lp(⌦)), we deduce that {f(un)}is bounded in Lq(⌧,T;Lq(⌦)). Thus, we deduce that there exist a function u2L1(⌧,T;L2(⌦))\L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)), ⇠12Lq(⌧,T;Lq(⌦)), ⇠22L2(⌧,T;H1(⌦)) and a subsequence of {un}(relabeled the same) such that 8 > > > > > > < > > > > > > : un ⇤ *uweakly-star in L1(⌧,T;L2(⌦)), un*uweakly in L2(⌧,T;H1 0(⌦)), un*uweakly in Lp(⌧,T;Lp(⌦)), f(un)*⇠ 1weakly in Lq(⌧,T;Lq(⌦)), a(l(un))un*⇠ 2weakly in L2(⌧,T;H1(⌦)). (6.9) To prove that ⇠1=f(u)and⇠2=a(l(u))u, we will use similar arguments to the ones used in the proof of Proposition 4.16. Consider w2Vnfixed. Integrating in (6.6) between tand t+b,withb2(0,T⌧)andt2(⌧,T b), and using (6.8) and the H¨older inequality, we obtain (un(t+b)un(t),w) (1")MC1Zt+b t kun(s)k2kwk2ds+"Zt+b t kh(s)k⇤kwk2ds+Zt+b t |f(un(s))|q|w|pds b1/2kwk2h(1 ")MC1kunkL2(⌧,T;H1 0(⌦)) +"khkL2(⌧,T;H1(⌦))i +b1/p|w|pkf(un)kLq(⌧,T;Lq(⌦)). Since {un}is bounded in L2(⌧,T;H1 0(⌦)) and {f(un)}is bounded in Lq(⌧,T;Lq(⌦)), there exists C">0 such that (un(t+b)un(t),w)C"(b1/2+b1/p)(kwk2+|w|p). Then, taking in the previous inequality w=un(t+b)un(t), we have |un(t+b)un(t)|2 2C"(b1/2+b1/p)(kun(t+b)un(t)k2+|un(t+b)un(t)|p) a.e. t2(⌧,T b). Now, integrating between ⌧and Tb,wehave ZTb ⌧ |un(t+b)un(t)|2 2dt 2C"(b1/2+b1/p)✓ZT ⌧ kun(r)k2dr +ZT ⌧ |un(r)|pdr◆. Chapter 6. Nonlocal reaction-di↵usion equations without uniqueness 147 Then, applying the H¨older inequality in the previous expression, we obtain ZTb ⌧ |un(t+b)un(t)|2 2dt 2C"(b1/2+b1/p)⇣(T⌧)1/2kunkL2(⌧,T;H1 0(⌦)) +(T⌧)1/qkunkLp(⌧,T;Lp(⌦))⌘. As a result of the previous estimates, there exists C"(T)>0 such that ZTb ⌧ |un(t+b)un(t)|2 2dt C"(T)(b1/2+b1/p)8n18b2(0,T ⌧). Therefore, lim b!0sup nZTb ⌧ |un(t+b)un(t)|2 2dt =0.(6.10) In addition, taking into account that {un}is bounded in L1(⌧,T;L2(⌦)), it is not difficult to check that lim b!0sup n✓Z⌧+b ⌧ |un(t)|2 2dt +ZT Tb |un(t)|2 2dt◆=0.(6.11) Then, since the embedding H1 0(⌦),!L2(⌦) is compact and taking into account (6.10) and (6.11), applying [110, Theorem 13.2, p. 97] and [110, Remark 13.1, p. 100], we obtain that the sequence {un}is relatively compact in L2(⌧,T;L2(⌦)).From this, making use of [85, Lemme 1.3, p. 12] and arguing as in the proof of Theorem 2.4, we identify ⇠1and ⇠2in (6.9). Namely, it has f(un)*f(u)weaklyinLq(⌧,T;Lq(⌦)), (6.12) a(l(un))un*a(l(u))uweakly in L2(⌧,T;H1 0(⌦)).(6.13) Now, we are ready to prove (6.4). Consider fixed n,'2D(⌧,T)andw2Vn Then, integrating (6.6) between ⌧and T,forallµ>nwe obtain ZT ⌧ (uµ(t),w)'0(t)dt +(1")ZT ⌧ a(l(uµ(t)))huµ(t),wi'(t)dt =ZT ⌧ (f(uµ(t)),w)'(t)dt +"ZT ⌧ hh(t),wi'(t)dt. Taking limit when µ!1, making use of (6.9), (6.12) and (6.13), it yields ZT ⌧ (u(t),w)'0(t)dt +(1")ZT ⌧ a(l(u(t)))hu(t),wi'(t)dt =ZT ⌧ (f(u(t)),w)'(t)dt +"ZT ⌧ hh(t),wi'(t)dt for all w2H1 0(⌦)\Lp(⌦), since Sn2NVnis dense in H1 0(⌦)\Lp(⌦).Therefore, (6.4) holds. 148 6.2. Minimal pullback attractors in L2(⌦) In addition, as a result of the previous equality, @u @t(1 ")a(l(u))u=f(u)+"hin D0(⌧,T;H1(⌦) + Lq(⌦)), and thanks to the regularity of f(u), a(l(u))uand h,wededucethatu02 L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)).As a consequence, u2C([⌧,T]; L2(⌦)) and it makes complete sense to check that u(⌧)=u⌧. On the one hand, consider fixed n,'2H1(⌧,T)suchthat'(T)=0and'(⌧)6=0 and w2Vn.From(6.6),wededuceforallµ>n (u⌧,w)'(⌧)ZT ⌧ (uµ(t),w)'0(t)dt +(1")ZT ⌧ a(l(uµ(t)))((uµ(t),w))'(t)dt =ZT ⌧ (f(uµ(t)),w)'(t)dt +"ZT ⌧ hh(t),wi'(t)dt. Now, taking limit when µ!1, we obtain (u⌧,w)'(⌧)ZT ⌧ (u(t),w)'0(t)dt +(1")ZT ⌧ a(l(u(t)))((u(t),w))'(t)dt =ZT ⌧ (f(u(t)),w)'(t)dt +"ZT ⌧ hh(t),wi'(t)dt. (6.14) On the other hand, in light of (6.4) we have (u(⌧),w)'(⌧)ZT ⌧ (u(t),w)'0(t)dt +(1")ZT ⌧ a(l(u(t)))((u(t),w))'(t)dt =ZT ⌧ (f(u(t)),w)'(t)dt +"ZT ⌧ hh(t),wi'(t)dt. Then, comparing (6.14) with this last expression, (u⌧,w)'(⌧)=(u(⌧),w)'(⌧) holds. Therefore, since '(⌧)6=0and{wj}is a Hilbert basis of L2(⌦), we deduce that u(⌧)=u⌧. We have obtained a weak solution on an arbitrary finite time interval [⌧,T]. Now, we may repeat this argument on an interval of the form [T,T + 1], then on [T+1,T +2],etcetera. This way, concatenating these solutions we finally obtain a weak solution well-defined globally in time. 6.2 Minimal pullback attractors in L2(⌦) In this section, we want to study the long-time behaviour of the solutions to (P")in L2(⌦) making use of the results of pullback attractors analysed in Chapter 5. In what follows, "(⌧,u⌧)denotesthesetofweaksolutionsto(P")in[⌧,1)with initial datum u⌧2L2(⌦). Now, we define the multi-valued map U":R2 d⇥L2(⌦)!P(L2(⌦)) by U"(t, ⌧)u⌧={u(t):u2"(⌧,u⌧)}8u⌧2L2(⌦)8⌧t. Chapter 6. Nonlocal reaction-di↵usion equations without uniqueness 149 Firstly, we show that the multi-valued map U"is a strict multi-valued process. Roughly speaking, this is a consequence of the translation and concatenation properties of the weak solutions. We will show the proof for the sake of completeness. Lemma 6.3. Assume that the function a2C(R;R+)and (6.1) holds, "2[0,1), f2C(R)satisfies (6.2), h2L2 loc(R;H1(⌦)) and l2L2(⌦). Then, the multi-valued map U"is a strict multi-valued process on L2(⌦)for all "2[0,1). Proof. Consider "2[0,1) fixed. The multi-valued map U"is well-defined because every weak solution ubelongs to C([⌧,T]; L2(⌦)) (cf. Theorem 6.2). In addition, observe that U"(⌧,⌧)u⌧={u(⌧):u2"(⌧,u⌧)}.Therefore, according to Definition 5.1, to prove that U"is a multi-valued process we only need to check U"(t, ⌧)u⌧⇢U"(t, s)U"(s, ⌧)u⌧8⌧st8u⌧2L2(⌦).(6.15) Consider fixed (t, ⌧)2R2 dand u⌧2L2(⌦).Given 2U"(t, ⌧)u⌧,thereexists u2"(⌧,u⌧)suchthatu(t)=.Observe that when s⌧,u(s)2U"(s, ⌧)u⌧. Then, since U"(t, s)u(s)={z(t):z2"(s, u(s))},wehave =u(t)2U"(t, s)u(s)⇢U"(t, s)U"(s, ⌧)u⌧. Therefore, (6.15) holds. Finally, we will check that in fact the multi-valued process U"is strict. To do this, consider given 2U"(t, s)U"(s, ⌧)u⌧.Then, there exists a solution uto (P") such that u(t)=and u(s)=z(s), where zis another solution to (P")whichfulfils that z(⌧)=u⌧. Now, we define y(r):=(z(r)if⌧rs, u(r)ifsrt. First of all, we will show that the function yis a weak solution to (P"). Taking into account the regularity of zand u, it holds that y2Lp(⌧,T;Lp(⌦))\L2(⌧,T;H1 0(⌦)) \C([⌧,T]; L2(⌦)). Therefore, it makes complete sense to define its distributional derivative as follows y0(r):=(z0(r)if⌧r<s, u0(r)ifsrt. Now, fix v2H1 0(⌦)\Lp(⌦) and '2D(⌧,t). Then, it holds hhy0(r),'(r)iD0(⌧,t;H1(⌦)+Lq(⌦)),D(⌧,t),vi =h Zt ⌧ y(r)'0(r)dr, vi =Zt ⌧ (y(r),v)'0(r)dr =Zs ⌧ (z(r),v)'0(r)dr Zt s (u(r),v)'0(r)dr =Zs ⌧ hz0(r),vi'(r)dr +Zt s hu0(r),vi'(r)dr. 150 6.2. Minimal pullback attractors in L2(⌦) Then, y02L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)).Finally, we need to check that the function yfulfils (6.4), i.e. given '2D(⌧,t), we need to prove that Zt ⌧ (y(s),v)'0(s)ds +(1")Zt ⌧ a(l(y(s)))((y(s),v))'(s)ds (6.16) =Zt ⌧ (f(y(s)),v)'(s)ds +"Zt ⌧ hh(s),vi'(s)ds. Observe that if supp(')⇢(⌧,s),(6.16) holds since zis a weak solution to (P")in (⌧,s).Analogously, if supp(')⇢(s, t), it satisfies (6.16) because uis a weak solution to (P")in(s, t).Let us prove (6.16) when supp(')6⇢ (⌧,s)andsupp(')6⇢ (s, t). Consider '2D(⌧,t)fixed. On the one hand, since uis a weak solution to (P")in(s, t), ufulfils (6.4) in D0(s, t).Then, since '2D(⌧,t), we have (u(s),v)'(s)Zt s (u(r),v)'0(r)dr +(1")Zt s a(l(u(r)))((u(r),v))'(r)dr =Zt s (f(u(r)),v)'(r)dr +"Zt s hh(r),vi'(r)dr. Analogously, as zis a weak solution to (P")in(⌧,s), zsatisfies (6.4) in D0(⌧,s). Therefore, making use of the fact that '2D(⌧,t), it yields (z(s),v)'(s)Zs ⌧ (z(r),v)'0(r)dr +(1")Zs ⌧ a(l(z(r)))((z(r),v))'(r)dr =Zs ⌧ (f(z(r)),v)'(r)dr +"Zs ⌧ hh(r),vi'(r)dr. On the other hand, u(s)=z(s) holds. Then, taking this into account, (6.16) holds. Therefore, the multi-valued process U"is strict. Remark 6.4. When "=0,U0(t, ⌧)=S(t⌧)for all (t, ⌧)2R2 d, where Sis the multi-valued semiflow associated to the weak solutions of the autonomous problem (P0). In what follows, we also keep the notation 0for the set of solutions to (P0). The following result is crucial to show that the multi-valued process U"is upper semicontinuous with closed values for all "2[0,1).To prove this continuity result, we use the energy method applied in Propositions 2.15 and 3.16 to analyse the pullback asymptotic compactness. Proposition 6.5. Under the assumptions of Lemma 6.3, consider a sequence {un ⌧} ⇢L2(⌦)such that un ⌧!u⌧strongly in L2(⌦). Then for any sequence {un}with un2"(⌧,un ⌧)for all n1, there exist a subsequence of {un}(relabeled the same) and u2"(⌧,u⌧)such that un(t)!u(t)strongly in L2(⌦)8t⌧.(6.17) Chapter 6. Nonlocal reaction-di↵usion equations without uniqueness 151 Proof. Consider ⌧<Tfixed. In view of the energy equality and (6.1), we deduce 1 2 d dt|un(t)|2 2+(1")mkun(t)k2 2(f(un(t)),u n(t)) + "hh(t),u n(t)ia.e. t2(⌧,T). Then, bearing in mind (f(un(t)),u n(t)) |⌦|↵2|un(t)|p p, "hh(t),u n(t)i "2kh(t)k2 ⇤ 2(1 ")m+(1 ")m 2kun(t)k2 2, we have d dt|un(t)|2 2+(1")mkun(t)k2 2+2↵2|un(t)|p p2|⌦|+"2 (1 ")mkh(t)k2 ⇤a.e. t>⌧. Integrating between ⌧and t2(⌧,T], |un(t)|2 2+(1")mZt ⌧ kun(s)k2 2ds +2↵2Zt ⌧ |un(s)|p pds |un ⌧|2 2+2|⌦|(T⌧)+ "2 (1 ")mZT ⌧ kh(s)k2 ⇤ds. From the previous inequality, we obtain that the sequence {un}is bounded in L1(⌧,T;L2(⌦)) \L2(⌧,T;H1 0(⌦)) \Lp(⌧,T;Lp(⌦)). Taking this into account together with the fact that each un2C([⌧,T]; L2(⌦)), we deduce that there exists a positive constant C1>0 such that |un(t)|2C18t2[⌧,T]8n1. Now, since the function a2C(R;R+)andl2L2(⌦), there exists a positive constant MC1>0suchthat a(l(un(t))) MC18t2[⌧,T]8n1. Therefore, as the sequence {un}is bounded in L2(⌧,T;H1 0(⌦)),we deduce that the sequence {a(l(un))un}is bounded in L2(⌧,T;H1(⌦)). In addition, the sequence {f(un)}is bounded in Lq(⌧,T;Lq(⌦)), thanks to (6.3) and the boundedness of {un}in Lp(⌧,T;Lp(⌦)).As a consequence, the sequence {(un)0}is bounded in L2(⌧,T;H1(⌦))+Lq(⌧,T;Lq(⌦)).Then, applying the Aubin-Lions lemma, there exist a subsequence of {un}(relabeled the same) and an element u2L1(⌧,T;L2(⌦))\ L2(⌧,T;H1 0(⌦))\Lp(⌧,T;Lp(⌦)) with u02L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)), such that8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : un⇤ *uweakly-star in L1(⌧,T;L2(⌦)), un*uweakly in L2(⌧,T;H1 0(⌦)), un*uweakly in Lp(⌧,T;Lp(⌦)), un!ustrongly in L2(⌧,T;L2(⌦)), un(s)!u(s) strongly in L2(⌦) a.e. (⌧,T), (un)0*u 0weakly in L2(⌧,T;H1(⌦)) + Lq(⌧,T;Lq(⌦)), f(un)*f(u)weaklyinLq(⌧,T;Lq(⌦)), a(l(un))un*a(l(u))uweakly in L2(⌧,T;H1(⌦)), (6.18) 152 6.2. Minimal pullback attractors in L2(⌦) where the limits of the last two convergences have been obtained applying [85, Lemme 1.3, p. 12], as done in the proof of Theorem 2.4. From (6.18), we deduce that ufulfils (6.4) in the interval (⌧,T).Moreover, since u2C([⌧,T]; L2(⌦)), a similar argument to the one used in the proof of Theorem 6.2 yields u(⌧)=u⌧.Therefore,u2"(⌧,u⌧). Finally, we will show the convergence (6.17). First of all, we have that the sequence {un}is equicontinuous in H1(⌦) + Lq(⌦) on [⌧,T], thanks to the boundedness of {(un)0}in L2(⌧,T;H1(⌦))+Lq(⌧,T;Lq(⌦)). Then, since the sequence {un} is bounded in C([⌧,T]; L2(⌦)) and the embedding L2(⌦),!H1(⌦)+Lq(⌦) is compact, the Arzela-Ascoli theorem implies (for another subsequence, relabeled again the same) the following convergence un!ustrongly in C([⌧,T]; H1(⌦) + Lq(⌦)).(6.19) Now, making use of the boundedness of {un}in C([⌧,T]; L2(⌦)),we deduce un(t)*u(t)weaklyinL2(⌦)8t2[⌧,T],(6.20) where we have used (6.19) to identify the weak limit. On the other hand, making use of (6.5), the following estimate holds |z(t)|2 2|z(s)|2 2+2|⌦|(ts)+ "2 2(1 ")mZt s kh(✓)k2 ⇤d✓8⌧stT, (6.21) with zreplaced by either uor any un. Now, we define the following functions Jn(t)=|un(t)|2 22|⌦|t"2 2(1 ")mZt ⌧ kh(r)k2 ⇤dr, J(t)=|u(t)|2 22|⌦|t"2 2(1 ")mZt ⌧ kh(r)k2 ⇤dr. As a result of the regularity of uand all un, the functions Jand all Jnare continuous on [⌧,T].Further, it is not difficult to check using (6.21) that these functions are non-increasing on [⌧,T].In addition, from (6.18), we deduce Jn(t)!J(t)a.e.t2(⌧,T). In fact, it can be proved Jn(t)!J(t)8t2[⌧,T].(6.22) To do this, consider t02(⌧,T]fixed. Let{tm}m1⇢(⌧,T)beasequencesuch that Jn(tm)!J(tm)forallm1andtm"t0.Now, fix ">0. Then, there exist m(")1andn(")1 such that |J(tm)J(t0)|<" 28mm("), |Jn(tm("))J(tm("))|<" 28nn("). Chapter 6. Nonlocal reaction-di↵usion equations without uniqueness 153 Taking this into account together with the non-increasing character of all Jnon [⌧,T], we obtain Jn(t0)J(t0)=Jn(t0)Jn(tm("))+Jn(tm("))J(tm("))+J(tm("))J(t0) |Jn(tm("))J(tm("))|+|J(tm("))J(t0)| <"8nn("). Then, bearing in mind the definitions of Jand Jn,wededuce lim sup n!1 |un(t0)|2 2|u(t0)|2 2. From this and (6.20) in t=t0, not only does (6.22) hold, but also (6.17) in [⌧,T]. Successive repetitions of this procedure in [⌧,T +1],[⌧,T +2],and so on, and a diagonal argument, yield (6.17) for all t⌧for a suitable subsequence. The following result establishes that the multi-valued process U"is upper-semi- continuous with closed values for all "2[0,1). Proposition 6.6. Under the assumptions of Lemma 6.3, the multi-valued process U"is upper semicontinuous with closed values for all "2[0,1). Proof. Fix "2[0,1). Firstly, we will show that the multi-valued process U"is upper semicontinuous. To prove it, we argue by contradiction. Suppose that there exist t⌧,u⌧2L2(⌦), a neighbourhood Nof U"(t, ⌧)u⌧,andasequence{yn}such that yn2U"(t, ⌧)un ⌧, where un ⌧!u⌧in L2(⌦) and yn62 Nfor all n1. Observe that since yn2U"(t, ⌧)un ⌧,thereexistsun2"(⌧,un ⌧)suchthatun(t)=yn. In addition, making use of Proposition 6.5, since un ⌧!u⌧strongly in L2(⌦), there exist a subsequence of {un}(relabeled the same) and u2"(⌧,u⌧) such that (6.17) holds. Therefore, there exists a subsequence of {yn}(relabeled the same) such that yn!u(t) strongly in L2(⌦), which contradicts the fact that yn62 Nfor all n2N. Then, the multi-valued process U"is upper semicontinuous. Finally, we will show that the multi-valued process U"has closed values, i.e. the set U"(t, ⌧)u⌧is closed in L2(⌦) for any u⌧2L2(⌦) and (t, ⌧)2R2 d. Fix (t, ⌧)2R2 d and u⌧2L2(⌦). Consider a sequence {un(t)}⇢U"(t, ⌧)u⌧converging strongly in L2(⌦). Then, un(⌧)=u⌧for all n2N.Since un(⌧)!u⌧strongly in L2(⌦), making use of Proposition 6.5, we deduce that there exists a subsequence of {un(t)} (relabeled the same) such that un(t)!u(t)2U"(t, ⌧)u⌧strongly in L2(⌦). This concludes the proof. The next result will be used to define a universe in P(L2(⌦)) that will be appropriate for our purposes. The idea of the proof is close to those of Lemmas 2.11 and 3.12. Proposition 6.7. Under the assumptions of Lemma 6.3, if u⌧2L2(⌦), then every solution uto (P")fulfils |u(t)|2 2eµ"(t⌧)|u⌧|2 2+2|⌦| µ" +"2eµ"t 2(1 ")m1 1µ"Zt ⌧ eµ"skh(s)k2 ⇤ds 8t⌧ (6.23)