Pullback attractors for a reaction-diffusion equation in a general nonempty open subset of R^N with non-autonomous forcing term in H^{−1}

Pullback attractors for a reaction-diffusion equation in a general nonempty open subset of RNwith non-autonomous forcing term in H−1 Mar´ıa Anguiano Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla (Spain) e-mail: [email protected] Abstract The existence of minimal pullback attractors in L2(Ω) for a non-autonomous reaction-diffusion equation, in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition, is proved in this paper, when the domain Ω is a general nonempty open subset of RN, and h∈ L2 loc(R;H−1(Ω)). The main concept used in the proof is the asymptotic compactness of the process generated by the problem. The relation among these families is also discussed. Keywords: Pullback attractor; asymptotic compactness; evolution process; non-autonomous reaction-diffusion equation 1 Introduction and setting of the problem In the entire paper, we assume that Ω ⊂RN, where N≥1, is a given nonempty open set. Let us consider the following problem for a non-autonomous reaction-diffusion equation with zero Dirichlet boundary condition in Ω,      ∂u ∂t −4u+κu =f(u) + h(t),in Ω ×(τ, +∞), u= 0,on ∂Ω×(τ, +∞), u(x, τ) = uτ(x), x ∈Ω, (1) where κ > 0, τ∈R,uτ∈L2(Ω), h∈L2 loc(R;H−1(Ω)) and f∈C(R) satisfies that there exist constants α1>0, α2>0, l > 0, and p > 2 such that −α1|s|p≤f(s)s≤ −α2|s|p, (2) (f(s)−f(r))(s−r)≤l(s−r)2∀r, s ∈R. (3) 1 Using (2), it follows that |f(s)| ≤ α1|s|p−1∀s∈R. (4) The aim of this paper is to show the existence of minimal pullback attractors in the phase space L2(Ω) for the problem (1) in the case of open domains. This, and the facts that the non-autonomous term hbelongs to the space L2 loc(R;H−1(Ω)) and the nonlinear term fsatisfies (3) with l > 0 , are the main novelties of our problem. The existence of the attractor for dissipative evolution equations has always relied on some kind of compactness of the process generated by such equations. Usually, the compactness is obtained through some regularization property of such equations together with the compact imbedding of the relevant Sobolev spaces. This approach is suitable only for bounded domains since Sobolev imbeddings are no longer compact otherwise. Our aim here is to avoid weighted spaces by exploiting the energy equation valid for the problem (1) in order to obtain the so-called asymptotic compactness of the process. The concept of asymptotic compactness was already used in the non-autonomous case (see [Caraballo et al.(2006a)] and [Caraballo et al.(2006b)]), and was previously used in [Rosa(1998)] for the autonomous case. Due to the non-autonomous character of our problem in this paper, we have to use an appropriate framework. Being possible to choose amongst several theories (skew-product flows, uniform attractors, trajectory attractors, pullback attractors) we will use the theory of pullback attractors since this allows for more generality in the non-autonomous terms (see [Anguiano et al.(2010)], [Caraballo et al.(2006a)], [Caraballo et al.(2006b)], [Mar´ın-Rubio & Real(2009)], [Mar´ın-Rubio & Real(2010)] for some results concerning pullback attractors and several reasons justifying the interest of using this theory). It is also worth mentioning that our problem has received much attention over the last years, as we will recall now. When Ω is bounded and h∈L2 loc(R;L2(Ω)) and is translation bounded, the existence of a pullback attractor in the space H1 0(Ω) is proved in [Song & Wu(2007)]. In [Wang & Zhong(2008)], the existence of pullback attractor in H1 0(Ω) is shown for a bounded domain and for h∈L2 loc(R;L2(Ω)). For a bounded domain Ω, and a translation bounded function h∈L2 loc(R;L2(Ω)), the existence of a uniform attractor in Lp(Ω) is demonstrated in [Song & Zhong(2008)]. When Ω is unbounded, the reader can find similar results for several variants of our model in [Morillas & Valero(2005)], [Prizzi(2003)] and [Sun & Zhong, 2005] for the autonomous case, and in [Wang et al.(2007)] for the non-autonomous case where h∈L2 loc(R;L2(Ω)). When Ω is not necessarily bounded but satisfying the Poincar´e inequality, h∈ L2 loc(R;H−1(Ω)) and fsatisfies (2) and (3) with l= 0, the existence of a pullback attractor in L2(Ω) is proved in [Anguiano et al.(2010)]. Our paper continues the line of investigation started in [Anguiano et al.(2010)]. But the fact that lis a positive constant in our model, implies that the techniques 2 previously used in [Anguiano et al.(2010)] do not work in our case. Therefore, we use a different technique which allow us to obtain a more general result (namely Theorem 21). We will provide in this paper a sufficient condition ensuring the existence of minimal pullback attractors in L2(Ω) when the domain is a general nonempty open subset of RN,h∈L2 loc(R;H−1(Ω)) and fsatisfies (2) and (3). A case that has not been considered in the literature yet, as far as we know. The existence of a pullback attractor begs a number of questions: the structure of the attractor, dimension estimates or the dynamics on the attractor, among others. We would like to investigate in a future work some of these questions, since the focus of this one is on the existence of minimal pullback attractors. The structure of the paper is as follows. In Section 2 we give a weak formulation of the problem, the concept of weak solution, and establish the existence and uniqueness of solution using the monotonicity method. A continuous dependence result with respect to initial data, which is the main key for the asymptotic compactness we will require later, is addressed in Section 3. There we use an energy method that strengthens the energy equality satisfied by the solutions. A brief recall on abstract results about the existence of minimal pullback attractors is given in Section 4. In Section 5, the main goals of proving the existence of different families of pullback attractors for different universes, and the relation among them under certain suitable assumption, are finally established. 2 Existence and uniqueness of solution We state in this section a result on the existence and uniqueness of solution of problem (1). Instead of working directly with our equation, we will apply a general result which is a slight modification of Theorem 1.4, Chapter 2 in [Lions(1969)]. By |·|,k·kH−1(Ω) and k·kLp(Ω) we denote the norms in the spaces L2(Ω), H−1(Ω) and Lp(Ω), respectively. By |∇·|2+κ|·|21/2we denote the norm in the space H1 0(Ω). We will use (·,·) to denote the scalar product in L2(Ω) or [L2(Ω)]N, and h·,·i to denote either the duality product between H−1(Ω) and H1 0(Ω) or between Lp0(Ω) and Lp(Ω), where p0=p p−1is the conjugate exponent of p. Definition 1 A weak solution of (1) is a function u, satisfying u∈C([τ, ∞); L2(Ω)),(5) u∈L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∀T > τ, (6)    d dt(u(t), v)+(∇u(t),∇v) + κ(u(t), v) = hf(u(t)), vi+hh(t), vi in D0(τ, ∞), for all v∈H1 0(Ω) ∩Lp(Ω), (7) 3 u(τ) = uτ.(8) Theorem 2 Assume that κ > 0,f∈C(R)satisfies (2) and (3), and h∈L2 loc(R; H−1(Ω)). Then, for all τ∈R,uτ∈L2(Ω), there exists a unique solution u(t) = u(t;τ, uτ)of the problem (1). Moreover, this solution satisfies the energy equality 1 2 d dt|u(t)|2+|∇u(t)|2+κ|u(t)|2=hf(u(t)), u(t)i+hh(t), u(t)i,a.e. t > τ. (9) Proof. The proof of this result is standard. For the sake of completeness, we give a sketch of a proof. Let us consider the spaces H=L2(Ω), V1=H1 0(Ω) and V2=Lp(Ω) ∩L2(Ω) with p > 2. Denote V0 1=H−1(Ω) and V0 2=Lp0(Ω) + L2(Ω). Recall that |·| denotes the norm in H, by k·k1=|∇·|2+κ|·|21/2we will denote the norm in V1, and by k·k2=k·kLp(Ω) +|·| the norm in V2. Now, we define a continuous symmetric linear operator A1:V1→V0 1, given by hA1(v), wi= (∇v, ∇w) + κ(v, w),∀v, w ∈H1 0(Ω). Let us denote A2(u) = −f(u), h1(t) = h(t), h2(t) = 0. From (4) one deduces that A2:V2→V0 2. On the other hand, we have that h1∈ L2 loc(R;V0 1). With this notation, and denoting V=∩2 i=1Vi, p1= 2, p2=p, one has that (5)–(8) is equivalent to u∈C([τ, ∞); H), u ∈ 2 \ i=1 Lpi(τ, T ;Vi),for all T > τ, (10) u0(t) + 2 X i=1 Ai(u(t)) = h(t) in D0(τ, ∞;V0),(11) u(τ) = uτ.(12) Applying a slight modification of [Lions(1969), Ch.2,Th.1.4], it is not difficult to see that problem (10)–(12) has a unique solution. Moreover, usatisfies the energy equality 1 2 d dt |u(t)|2+ 2 X i=1hAi(u(t)), u(t)ii= (h(t), u(t)) a.e. t > τ, where h·,·iidenotes the duality product between V0 iand Vi. This last equality turns out to be just (9). 4 3 A continuous dependence result In this section, we give a result on continuous dependence of the solutions of (1) with respect to the initial datum uτ. Theorem 3 Assume that the assumptions of Theorem 2 are satisfied. Let {u(n) τ}n≥1 ⊂L2(Ω) be a sequence such that u(n) τ* uτweakly in L2(Ω).(13) Let us denote u(n)=u(·;τ, u(n) τ)and u=u(·;τ, uτ)the corresponding weak solutions of (1). Then, for all T > τ, u(n)* u weakly in L2(τ, T ;H1 0(Ω)),(14) u(n)* u weakly in Lp(τ, T ;Lp(Ω)), f(u(n))* f(u)weakly in Lp0(τ, T ;Lp0(Ω)), u(n)(t)* u(t)weakly in L2(Ω),for all t > τ. (15) If Ωis a bounded set, then for all T > τ, u(n)→ustrongly in L2(τ, T ;L2(Ω)),(16) u(n)(t)→u(t)strongly in L2(Ω),for all t > τ. (17) Proof. For all but last of the above convergences we argue similarly to the proof of Proposition 4.1 in [Anguiano et al.(2010)]. For the last convergence, we use an energy method that strengthens the energy equality satisfied by the solutions. From (16), we deduce that from every subsequence of u(n)we can extract a subsequence that we will denote by {uν}, such that |uν(t)|→|u(t)|a.e. in (τ, T ) . (18) Let us define Jν(t) = 1 2|uν(t)|2−Zt τhh(s), uν(s)ids, and J(t) = 1 2|u(t)|2−Zt τhh(s), u(s)ids, for all t≥τ. It is clear that Jνand Jare continuous functions. Also, from (14) and (18) we see that Jν(t)→J(t)a.e. t ∈(τ, T ) as ν→ ∞.(19) On the other hand, taking into account (9) and (2), we obtain 1 2 d dt |uν(t)|2≤ hh(t), uν(t)i,t>τ. 5 Thus, for every ν, the function Jνis a non-increasing function of t. We are now in position to show that Jν(t)→J(t) strongly for all t∈(τ, T ).(20) Let t∈(τ, T ) and ε > 0 be fixed. From (19) and the continuity of J, we can take t0> t and t00 < t such that Jν(t0)→J(t0) strongly as ν→ ∞,(21) Jν(t00)→J(t00) strongly as ν→ ∞,(22) J(t00)−J(t)≤ε, (23) and J(t)−J(t0)≤ε. (24) As Jνis a non-increasing function of t, we obtain Jν(t0)−Jν(t)≤0, (25) and Jν(t00)−Jν(t)≥0, (26) for every ν. Using (23) and (26) we have Jν(t)−J(t) = Jν(t)−Jν(t00) + Jν(t00)−J(t00) (27) +J(t00)−J(t) ≤Jν(t00)−J(t00)+ε. Analogously, using (24) and (25) we obtain J(t)−Jν(t) = J(t)−J(t0) + J(t0)−Jν(t0) +Jν(t0)−Jν(t) ≤J(t0)−Jν(t0)+ε. (28) From (21), (22), (27) and (28), we have lim sup ν→∞ |J(t)−Jν(t)| ≤ ε, (29) and therefore, as ε > 0 is arbitrary, (20) follows from (29). Thanks to (29), and taking into account (14), we deduce that |uν(t)|→|u(t)|strongly ∀t∈(τ, T ) , and then, by (15), we obtain uν(t)→u(t) strongly in L2(Ω) ∀t∈(τ, T ) . Then from a standard contradiction argument combined with the fact that T > τ has been taken arbitrarily, we deduce that (17) holds. 6 4 Abstract results on minimal pullback attractors In this section we remember some abstract results on pullback attractors theory. We present a resume of some results on the existence of minimal pullback attractors obtained in [Garc´ıa-Luengo et al.(2012)] (see also [Caraballo et al.(2006a)], [Caraballo et al.(2006b)], [Mar´ın-Rubio & Real(2009)]). In particular, we consider the process Ubeing closed (see below Definition 4). Consider a given metric space (X, dX),and let us denote R2 d={(t, τ)∈R2:τ≤ t}. A process on Xis a mapping Usuch that R2 d×X3(t, τ, x)7→ U(t, τ)x∈X with U(τ, τ)x=xfor any (τ, x)∈R×X, and U(t, r)(U(r, τ)x) = U(t, τ)xfor any τ≤r≤tand all x∈X. Definition 4 Let Ube a process on X. a) Uis said to be continuous if for any pair τ≤t, the mapping U(t, τ) : X→X is continuous. b) Uis said to be closed if for any τ≤t, and any sequence {xn} ⊂ X, if xn→x∈Xand U(t, τ)xn→y∈X, then U(t, τ)x=y. Remark 5 It is clear that every continuous process is closed. More generally, every strong-weak continuous process (see [Mar´ın-Rubio & Real(2009)] for the definition) is a closed process. Let us denote P(X) the family of all nonempty subsets of X, and consider a family of nonempty sets b D0={D0(t) : t∈R}⊂P(X) [observe that we do not require any additional condition on these sets as compactness or boundedness]. Definition 6 We say that a process Uon Xis pullback b D0-asymptotically compact if for any t∈Rand any sequences {τn} ⊂ (−∞, t]and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D0(τn)for all n, the sequence {U(t, τn)xn}is relatively compact in X. Let be given Da nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X).The class Dwill be called a universe in P(X). Definition 7 It is said that b D0={D0(t) : t∈R}⊂P(X)is pullback D−absorbing for the process Uon Xif for any t∈Rand any b D∈ D,there exists a τ0(t, b D)≤t such that U(t, τ)D(τ)⊂D0(t)for all τ≤τ0(t, b D). Observe that in the definition above b D0does not belong necessarily to the class D. Definition 8 A process Uon Xis said to be pullback D−asymptotically compact if it is b D-asymptotically compact for any b D∈ D,i.e. if for any t∈R,any b D∈ D,and any sequences {τn} ⊂ (−∞, t]and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D(τn) for all n, the sequence {U(t, τn)xn}is relatively compact in X. 7 Denote Λ( b D0, t) := \ s≤t[ τ≤s U(t, τ)D0(τ) X for all t∈R, where {···}Xis the closure in X. We denote by distX(O1,O2) the Hausdorff semi-distance in Xbetween two sets O1and O2, defined as distX(O1,O2) = sup x∈O1 inf y∈O2 dX(x, y) for O1,O2⊂X. We have the following result (cf. [Garc´ıa-Luengo et al.(2012)]) on existence of minimal pullback attractors. Theorem 9 Consider a closed process U:R2 d×X→X, a universe Din P(X), and a family b D0={D0(t) : t∈R} ⊂ P(X)which is pullback D−absorbing for U, and assume also that Uis pullback b D0−asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = [ b D∈D Λ( b D, t) X t∈R, has the following properties: (a) for any t∈R,the set AD(t)is a nonempty compact subset of X, and AD(t)⊂Λ( b D0, t), (b) ADis pullback D−attracting, i.e. lim τ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D, t ∈R, (c) ADis invariant, i.e. U(t, τ)AD(τ) = AD(t)for all τ≤t, (d) if b D0∈ D,then AD(t) = Λ( b D0, t)⊂D0(t)X,for all t∈R. The family ADis minimal in the sense that if b C={C(t) : t∈R} ⊂ P(X)is a family of closed sets such that for any b D={D(t) : t∈R}∈D, lim τ→−∞ distX(U(t, τ)D(τ), C(t)) = 0, then AD(t)⊂C(t). 8 Remark 10 Under the assumptions of Theorem 9, the family ADis called the minimal pullback D−attractor for the process U. If AD∈ D,then it is the unique family of closed subsets in Dthat satisfies (b)–(c). A sufficient condition for AD∈ D is to have that b D0∈ D, the set D0(t)is closed for all t∈R,and the family Dis inclusion-closed (i.e. if b D∈ D,and b D0={D0(t) : t∈R}⊂P(X)with D0(t)⊂D(t)for all t, then b D0∈ D). We will denote DX Fthe universe of fixed nonempty bounded subsets of X, i.e. the class of all families b Dof the form b D={D(t) = D:t∈R}with Da fixed nonempty bounded subset of X. In the particular case of the universe DX F, the corresponding minimal pullback DX F−attractor for the process Uis the pullback attractor defined by Crauel, Debussche, and Flandoli, [Crauel et al.(1997), Th.1.1, p.311], and will be denoted ADX F. Now, it is easy to conclude the following result. Corollary 11 Under the assumptions of Theorem 9, if the universe Dcontains the universe DX F,then both attractors, ADX Fand AD,exist, and the following relation holds: ADX F(t)⊂ AD(t)for all t∈R. Remark 12 It can be proved (see [Mar´ın-Rubio & Real(2009)]) that, under the assumptions of the preceding corollary, if, moreover, b D0∈ D,and for some T∈Rthe set ∪t≤TD0(t)is a bounded subset of X, then ADX F(t) = AD(t)for all t≤T. 5 Existence of pullback attractors Now, by the previous results, we are able to define correctly a process Uon L2(Ω) associated to (1), and to obtain the existence of minimal pullback attractors. From the uniqueness of solution to problem (1), we can define a process in L2(Ω). In addition, it is easy to prove that the process is continuous in L2(Ω). Proposition 13 Assume that the assumptions of Theorem 2 are satisfied. Then, the family of maps U(t, τ) : L2(Ω) →L2(Ω), with τ≤t, given by U(t, τ)uτ=u(t),(30) where u=u(·;τ, uτ)is the unique weak solution of (1), defines a continuous process on L2(Ω). 9 If moreover hsatisfies sup s≤0 e−κs N X i=0 Zs −∞ eκθ |hi(θ)|2dθ!<+∞,(50) then ADL2(Ω) F (t) = ADκ(t)for all t∈R.(51) Proof. All but the last results are consequences of Theorem 9 and Corollary 11. Finally, (51) follows from (50) and Remark 12, taking into account the expression R(t) given in Corollary 18. Acknowledgments I would like to thank the referees for their comments and suggestions. Also, I would like to thank my father, Julio, for his support in this stage of my career. References [Anguiano et al.(2010)] Anguiano, M., Caraballo, T. & Real, J. [2010] “Existence of pullback attractor for a reaction-diffusion equation in some unbounded domains with non-autonomous forcing term in H−1”, International Journal of Bifurcation and Chaos,20, No. 9, 2645-2656. [Brezis(1983)] Brezis, H. [1983] Analyse Fonctionnelle. Thorie et Applications, Masson, Paris. [Caraballo et al.(2006a)] Caraballo, T., Lukaszewicz, G. & Real, J. [2006a] “Pullback attractors for asymptotically compact non-autonomous dynamical systems”, Nonlinear Anal. 64, 484–498. [Caraballo et al.(2006b)] Caraballo, T., Lukaszewicz, G. & Real, J. [2006b] “Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains”, C. R. Math. Acad. Sci. Paris,342, 263–268. [Crauel et al.(1997)] Crauel, H., Debussche, A. & Flandoli, F. [1997] “Random attractors”, J. Dynam. Differential Equations,9, 307–341. [Garc´ıa-Luengo et al.(2012)] Garc´ıa-Luengo, J., Mar´ın-Rubio, P. & Real, J. [2012] “Pullback attractors in Vfor non-autonomous 2D-Navier-Stokes equations and their tempered behaviour”, J. Differential Equations,252, No. 8, 4333-4356. [Lions(1969)] Lions, J.L. [1969] Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non lin`eaires, Dunod. 16 [Mar´ın-Rubio & Real(2009)] Mar´ın-Rubio, P. & Real, J. [2009] “On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems”, Nonlinear Anal.,71, 3956–3963. [Mar´ın-Rubio & Real(2010)] Mar´ın-Rubio, P. & Real, J. [2010] “Pullback Attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators”, Discrete Contin. Dyn. Syst.,26, 989–1006. [Morillas & Valero(2005)] Morillas, F. & Valero, J. [2005] “Attractors for reactiondiffusion equations in RNwith continuous nonlinearity”, Asymptotic Analysis, 44, 111-130. [Prizzi(2003)] Prizzi, M. [2003] “A remark on reaction-diffusion equations in unbounded domains”, Discrete Contin. Dyn. Syst.,9, 281-286. [Rosa(1998)] Rosa, R. [1998] “The Global Attractor for the 2D Navier-Stokes flow on some unbounded domains” Nonlinear Anal. 32, no. 1, 71-85. [Song & Wu(2007)] Song, H., & Wu, H. [2007] “Pullback attractors of nonautonomous reaction-diffusion equations,” J. Math. Anal. Appl. 325, 1200-1215. [Song & Zhong(2008)] Song, H. & Zhong, C.K. [2008] “Attractors of nonautonomous reaction-diffusion equations in Lp”, Nonlinear Analysis,68, 1890- 1897. [Sun & Zhong, 2005] Sun, C. Y., & Zhong, C. K. [2005] “Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains,” Nonlinear Analysis 63, 49-65. [Wang et al.(2007)] Wang, Y., Wang, L., & Zhao, W. [2007] “Pullback attractors of nonautonomous reaction-diffusion equations in unbounded domains,” J. Math. Anal. Appl.336, 330-347. [Wang & Zhong(2008)] Wang, Y., & Zhong, C. [2008] “On the existence of pullback attractors for non-autonomous reaction-difusion equations,” Dynamical Systems, 23, 1-16. 17