Asymptotic behavior of nonlocal partial differential equations with long time memory

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 ASYMPTOTIC BEHAVIOR OF NONLOCAL PARTIAL DIFFERENTIAL EQUATIONS WITH LONG TIME MEMORY Jiaohui Xu, Tom´ as Caraballo Departmento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain Jos´ e Valero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche, Avda. de la Universidad, s/n, 03202-Elche, Spain (Communicated by the associate editor name) Dedicated to Georg Hetzer on occasion of his 75th birthday Abstract. In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70’s. Thanks to this Dafermos transformation, the original problem with memory is transformed into a nondelay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors to the transformed problem are provided. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem, it is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem. Keywords: Nonlocal partial differential equations, Long time memory, Dafermos transformation, Global attractors AMS subject classifications: 35B40, 35K05, 37L30, 45K05, The research has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andaluc´ıa (Consejer´ıa de Econom´ıa y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509. 1 2 NONLOCAL PDE WITH LONG TIME MEMORY 1. Introduction. Many physical phenomena are properly described by (systems of) partial differential equations whose dynamics is influenced by the past history of one or more variables (see [1, 6, 2, 7, 11, 13] and the references therein). Also, in recent decades, nonlocal problems have been investigated with great interest due to their wide applications in the real world (see, [3, 4, 5, 6] and the references therein). There is a significant amount of literature studying partial differential equations with long time memory. For example, the authors in [6] introduced the following semilinear problem describing the heat flow in a rigid, isotropic, homogeneous heat conductor with linear memory,          c0∂tθ−k0∆θ−Zt −∞ k(t−s)∆θ(s)ds +g(θ) = h, θ(x, t)=0, θ(x, t) = θ0(x, t), in Ω ×R+, on ∂Ω×R+, in Ω ×(−∞,0], (1) where Ω ⊂RNis a fixed bounded domain with regular boundary occupied by a rigid heat conductor, θ: Ω ×R→Ris the temperature variation field relative to the equilibrium reference value, k:R+→Ris the heat flux memory kernel, the constants c0and k0denote the specific heat and the instantaneous conductivity, respectively (R+denotes the open interval (0,+∞)). To handle (1) successfully, the authors considered this problem as a non-delay one by making the past history of θfrom −∞ to 0−be part of the forcing term given by the causal function f, which is defined by, f(x, t) = h(x, t) + Z0 −∞ k(t−s)∆θ(x, s)ds, x ∈Ω, t ≥0. In this way, (1) becomes an initial value problem without delay or memory,          c0∂tθ−k0∆θ−Zt 0 k(t−s)∆θ(s)ds +g(θ) = f, θ(x, t)=0, θ(x, 0) = θ0(x, 0), in Ω ×R+, on ∂Ω×R+, in Ω. (2) Amongst the many notable results regarding the nonlocal differential equations, it is worth mentioning the pioneer work [9], in which the authors examined one model of single-species dynamics incorporating nonlocal effects, comparing with the standard approach to model a single-species domain Ω of “Kolmogorov” type, ∂tu= ∆u+λug(u),in Ω, t > 0. If we take into account the following backgrounds: (i) a population in which individuals compete for a shared rapidly equilibrate resource; (ii) a population in which individuals communicate either visually or by chemical means, then the most straightforward way of introducing nonlocal effects is to consider, instead of g(u), a “crowding” effect of the form g(u, ¯u), where ¯u(x, t) = ZΩ G(x, y)u(y, t)dy, and G(x, y) is some reasonable kernel. Heuristically, Chipot et al. studied in [5] the behavior of a population of bacteria with nonlocal term a(RΩu) in a container. Later, Chipot et al. extended this term to a general nonlocal operator a(l(u)) (cf. [3, 4, 5]), where l∈ L(L2(Ω); R), for instance, if g∈L2(Ω), l(u) = lg(u) = ZΩ g(x)u(x)dx. NONLOCAL PDE WITH LONG TIME MEMORY 3 Inspired by the above work, the dynamics of the following non-autonomous nonlocal partial differential equations with delay and memory was investigated in [17] by using the Galerkin method and energy estimations,      ∂u ∂t −a(l(u))∆u=f(u) + h(t, ut), u= 0, uτ(x, t) = ϕ(x, t), in Ω ×(τ, ∞), on ∂Ω×R, in Ω ×(−ρ, 0], (3) where Ω ⊂RNis a bounded open set, τ∈R, function a∈C(R;R+) is locally Lipschitz, f∈C(R) and hcontains hereditary characteristics involving delays, and ut: (−∞,0] →Ris a segment of the solution given by ut(x, s) = u(x, t +s), s≤0, which essentially is the history of the solution up to time t. Moreover, 0 < ρ ≤ ∞, which implies that the authors considered both cases, bounded and unbounded delays, for this model. However, the technique applied in this paper to deal with memory terms is limited, since only exponential kernels can be contained in the delay terms. For instance, h(t, ut) = Rt −∞ k(t−s)∆u(x, s)ds, where k(·) is nonsingular and of sub-exponential type (e.g., k(t) = k1e−d0t,k1∈R, d0>0), for more details, see [1]. Whereas, this technique fails to deal with various important models with memory, whose kernels have singularities. Consequently, it is not surprising that very recently a new discussion has started concerning the following long time memory differential equation associated with nonlocal diffusion,          ∂u ∂t −a(l(u))∆u−Zt −∞ k(t−s)∆u(s)ds +f(u) = g, u(x, t) = 0, u(t+τ) = u0(t), in Ω ×(τ, ∞), on ∂Ω×R, in Ω ×(−∞,0], (4) where Ω ⊂RNis a fixed bounded domain with regular boundary, a∈C(R;R+) satisfies 0< m ≤a(r),∀r∈R,(5) k:R+→Ris the memory kernel, with or without singularities, whose properties will be specified later, g∈L2(Ω) which is independent of time. Notice that, thanks to a change of variable, the long time memory term in problem (4) can be interpreted as an infinite delay term, Zt −∞ k(t−s)∆u(x, s)ds =Z0 −∞ k(−s)∆u(x, t +s)ds =Z0 −∞ k(−s)∆ut(x, s)ds := h(ut).(6) In this way, it is obvious that we are working on an autonomous nonlocal partial differential equation. Problem (4) was analyzed in [1] when a(·) is a constant function (local problem) and the kernel kis of special type mentioned above. It is proved that the problem generates a dynamical system in the phase space L2 H1 0given by the measurable functions ϕ: (−∞,0] →H1 0(Ω), such that R0 −∞ eγskϕ(s)k2 H1 0 ds < +∞. Under the construction of this phase space, there exists a global attractor to this problem (in fact, the problem in [1] is non-autonomous and the attractor is of pullback type). Notice that, for this kind of delay problems, in which the initial value at zero may not be related to the values for negative times, the standard phase space to construct a dynamical system is the cartesian product L2(Ω) ×L2 H1 0. In such a way, for any initial values u0∈L2(Ω) and 4 NONLOCAL PDE WITH LONG TIME MEMORY ϕ∈L2 H1 0, there exists a unique solution to the following problem,              ∂u ∂t −a∆u−Zt −∞ k(t−s)∆u(s)ds +f(u) = g, u(x, t)=0, u(x, 0) = u0(x), u(x, t) = ϕ(x, t), in Ω ×(0,∞), on ∂Ω×R, in Ω, in Ω ×(−∞,0). (7) According to the regularity of solutions to the above equation, one can define a dynamical system S(t) : L2(Ω) ×L2 H1 0→L2(Ω) ×L2 H1 0by the relation S(t)(u0, ϕ) := (u(t; 0, u0, ϕ), ut(·;u0, ϕ)), where u(·; 0, u0, ϕ) denotes the solution of problem (7) (see [2] for more details on this set-up). We emphasize that the two components of the dynamical system are the current state of the solution and the past history up to present, respectively, what is sensible in a problem with delays or memory. By using this framework, the method in [1] can be successfully applied to prove the existence of attractors to problem (7) when kis of exponential type. As for much more realistic cases, an alternative method based on the Dafermos transformation can provide us some ideas to study the long time behavior of the problem. The idea to bypass problem (4) is to redefine gas g+Zτ −∞ k(t−s)∆u(x, s)ds, hence, the new external term gincludes the memory or history of the problem. Then, a way to associate a semigroup with such equation is to view the past history of u(in integrated form) as a new variable of the system, which will be ruled by a supplementary equation, ηt t=−ηt s+u. This idea goes back to the seventies, and it was first introduced by Dafermos [8] for linear viscoelasticity. We define the new variables, ut(x, s) = u(x, t −s), s ≥0, t ≥τ, and ηt(x, s) = Zs 0 ut(x, r)dr =Zt t−s u(x, r)dr, s ≥0, t ≥τ. (8) Assuming k(∞) = 0, a change of variable and a formal integration by parts yield Zt −∞ k(t−s)∆u(s)ds =−Z∞ 0 k0(s)∆ηt(s)ds. Here and in the sequel, the prime denotes derivation with respect to the variable s. Setting µ(s) = −k0(s), the original equation (4) turns into the following autonomous system without delay,                  ∂u ∂t −a(l(u))∆u−Z∞ 0 µ(s)∆ηt(s)ds +f(u) = g, in Ω ×(τ, ∞), ηt t(s) = −ηt s(s) + u(t),in Ω ×(τ, ∞)×R+, u(x, t) = ηt(x, s)=0,on ∂Ω×R×R+ u(x, τ) = u0(0),in Ω, ητ(x, s) = η0(s),in Ω ×R+. (9) NONLOCAL PDE WITH LONG TIME MEMORY 5 where, ηt sdenotes the distributional derivative of ηt(s) with respect to the internal variable s. It follows from the definition of ηt(x, s) (see (8)) that η0(s) = Zτ τ−s u(r)dr =Zτ τ−s u0(r−τ)dr =Z0 −s u0(r)dr, (10) which is the initial integrated past history of uwith vanishing boundary. Consequently, any solution to (4) is solution to (9) for the corresponding initial values (u0(0), η0) given by (10). It is worth emphasizing that problem (9) can be solved for arbitrary initial values (u0, η0) in a proper phase space L2(Ω) ×L2 µ(R+;H1 0(Ω)) (see Section 2), i.e., the second component ηtdoes not need to depend on u0. This allows us to construct a dynamical system in this phase space and prove the existence of global attractors. However, the transformed equation (9) is in fact a generalization of problem (4), which means, not every solution to equation (9) possesses a corresponding one to (4). In fact, both problems are equivalent if and only if the initial value η0belongs to a proper subset of L2 µ(R+;H1 0(Ω)), which coincides with the domain of the distributional derivative with respecto to s, denoted by D(T) (for more details, see [11]). In this paper, when we work on (9) assuming η0∈D(T), we will prove existence and uniqueness of solution to problem (9), and this also provides us the existence of solution to problem (4). However, when η0does not necessarily belong to D(T), we may not have any corresponding solution to the original problem. Once we construct the dynamical system in the phase space L2(Ω) ×L2 µ(R+;H1 0(Ω)), the existence of absorbing sets and an asymptotic compactness property ensuring the existence of global attractor for the problem (9) can be shown, but we are not able to prove that such an attractor exists in a more regular phase space (related to D(T) for the second component of the dynamical system), so that we could have the complete equivalent information for the initial problem (4). This is also the case in the papers dealing with local problems (see [6, 7, 11]). The content of this paper is as follows. In Section 2, we recall some preliminaries and notations which are necessary for our analysis. Section 3 is devoted to proving the existence and uniqueness of solution to problem (9) as well as some necessary regularity results and continuous dependence on the initial values. The main techniques are the Faedo-Galerkin scheme and some compactness results. The nonlocal term introduces some technical difficulties which are successfully overcome. Next, in Section 4, it is first proved the existence of absorbing sets in two phase spaces and the asymptotic compactness of the dynamical system, which ensure the existence of global attractors. Eventually, in Section 5, some conclusions and further comments for future investigations are included. 2. Preliminaries. The purpose of this section is to state assumptions on µand f, which allow us to prove properly the well-posedness and asymptotic behavior of problem (9). In addition, some notations will be introduced so that we can study our problem in suitable phase spaces. At last, we will establish several lemmas and corollaries for the supplementary equation. 2.1. Assumptions. Suppose the nonlinear term, f:R→R, is a polynomial of odd degree with positive leading coefficient, f(u) = 2p X k=1 f2p−kuk−1, p ∈N.(11) 6 NONLOCAL PDE WITH LONG TIME MEMORY Remark 1. (a) In fact, to prove the main results of this paper, we can consider, more generally, a continuously differentiable function fon Rsatisfying (i) |f(u)| ≤ k1(1 + |u|β), (ii) u·f(u)≥ −k2+k3|u|β+1, (iii) f0(u)≥ −k4, for some β > 0 and kj≥0, j= 1,2,3,4. The proofs of main results in this paper do not have significant changes with assumptions (i)-(iii) comparing to (11). (b) Comparing with assumptions (2.3)-(2.4) to nonlinear term f(u) in Section 2 in [17], here, (11) is the quite weaker condition which has less restrictions on f(u). We also want to emphasize if condition (11) were adopted in [17] instead of (2.3)-(2.4), the method to prove main results of [17] can not be used successfully. In view of the evolution problem (9), the variable µis required to verify the following hypotheses, (h1)µ∈C1(R+)∩L1(R+), µ(s)≥0, µ0(s)≤0, ∀s∈R+; (h2)µ0(s) + δµ(s)≤0, ∀s∈R+, for some δ > 0. Remark 2. (i) Restriction (h1)is equivalent to assuming k(s)is a bounded, positive, non-increasing, convex function of class C2vanishing at infinity. Moreover, from (h1)it easily follows that k(0) = Z∞ 0 µ(s)ds is finite and nonegative. (ii) Observe that, by the Gronwall inequality, (h2)implies µ(s)decays exponentially, i.e., µ(s)≤µ(s0)e−δ(s−s0),∀s≥s0>0.(12) 2.2. Notations. Throughout this manuscript, we make use of several notations introduced in what follows. Let | · |pdenotes the Lp-norm for p≥1, let (·,·) and | · | denote the L2-inner product and L2-norm, ((·,·)) and k·kdenote the H1 0-inner product and H1 0-norm, respectively. Recall that for every v∈H1 0(Ω), the Poincar´e inequality λ1(Ω)|v|2≤ kvk2 holds. If v∈H2(Ω) ∩H1 0(Ω), the Poincar´e and Young inequalities yield λ1(Ω)|∇v|2≤ |∆v|2, where λ1(Ω) is the first eigenvalue of −∆ with zero Dirichlet boundary conditions on Ω. In the sequel, unless otherwise specified, we write λ1instead of λ1(Ω) for simplicity. To simplify the presentation, from now on, the space L2(Ω) is denoted by H,H1 0(Ω) is denoted by Vand H1 0(Ω) ∩ H2(Ω) is denoted by D(A). As usual, V∗is the dual space of V, whose norm is denoted by k·k∗. In view of system (9) and (h1), it is necessary to introduce suitable Banach spaces, which aims at capturing the essence of the problem. Let L2 µ(R+;H) be the Hilbert space of functions w:R+→H endowed with the inner product, (w1, w2)µ=Z∞ 0 µ(s)(w1(s), w2(s))ds, and let | · |µdenote the corresponding norm. In a similar way, we introduce the inner products ((·,·))µ, (((·,·)))µand corresponding norms k·kµ,|||·|||µon L2 µ(R+;V), L2 µ(R+;D(A)), respectively. It follows then ((·,·))µ= (∇·,∇·)µand (((·,·)))µ= (∆·,∆·)µ. NONLOCAL PDE WITH LONG TIME MEMORY 7 We also define the Hilbert spaces, H=H×L2 µ(R+;V) and V=V×L2 µ(R+;D(A)), which are respectively endowed with inner products, (w1, w2)H= (w1, w2) + ((w1, w2))µ, and (w1, w2)V= ((w1, w2)) + (((w1, w2)))µ, where wi∈ H or V(i= 1,2). The norms induced on Hand Vare the so-called energy norm, which read, k(w1, w2)k2 H=|w1|2+Z∞ 0 µ(s)kw2(s)k2ds, and k(w1, w2)k2 V=kw1k2+Z∞ 0 µ(s)k∇w2(s)k2ds. Let Tbe the linear operator with domain D(T) = η(·)∈L2 µ(R+;V)|ηs(·)∈L2 µ(R+;V), η(0) = 0, defined by Tη=−ηs, η ∈D(T). Indeed, Tis the infinitesimal generator of the right-translation semigroup on L2 µ(R+;V), see [11]. Eventually, we introduce the following space E={η(·)∈L2 µ(R+;D(A)) |η(·)∈D(T),sup r≥1 r£η(r)<∞}. Here, £ηis the tail function of η(·), which is given by £η(r) = Z(0,1/r)∪(r,∞) µ(s)kη(s)k2ds, r ≥1. It is straightforward to see (cf. [10]) that Eis a Banach space endowed with the norm kηk2 E=kηk2 L2 µ(R+;D(A)) +kTηk2 L2 µ(R+;V)+ sup r≥1 r£η(r). At this point, an immediate generalization of a compactness result is presented in [15, Lemma 3.1]. Lemma 3. Let K ⊂ L2 µ(R+;V), satisfying sup η∈K hkηkL2 µ(R+;D(A)) +kηskL2 µ(R+;V∗)i<∞and lim r→∞ sup η∈K £η(r)= 0. Then Kis relatively compact in L2 µ(R+;V). At last, with standard notations, Dc(I;X) is the space of infinitely differentiable X-valued function with compact support in I⊂R, whose dual space is the distribution space on Iwith values in X∗(dual of X), denoted by D0(I;X∗). 8 NONLOCAL PDE WITH LONG TIME MEMORY 2.3. The representation formula and some useful lemmas. We recall the following definitions and properties of the supplementary equation, for more details, see [7, 11] and references therein. Assume that uis a given function belonging to L1(τ, T;V) for every T > τ. Then, for every η0∈L2 µ(R+;V), the Cauchy problem (d dt ηt(s) = Tηt(s) + u, t ≥τ, η0=η0, t < τ, s ≥0, has a unique solution η·∈C(τ, ∞;L2 µ(R+;V)), which has the explicit expression formula (see [11]), ηt(s) =        Zt t−s u(r)dr, 0< s ≤t, η0(s−t) + Zt 0 u(t−r)dr, s > t. (13) We next show estimates for the functions ηtbelonging to E, since they play an important role in the proof of the asymptotically compact theorem. Lemma 4. Let η0∈D(T), assume that ku(t)k ≤ ρ, for some ρ > 0and every t≥τ. Then kTηtk2 L2 µ(R+;V)≤e−δtkTη0k2 L2 µ(R+;V)+ρ2kµkL1. Lemma 5. Let η0∈D(T), assume that ku(t)k ≤ ρ, for some ρ > 0and t≥τ. Then sup r≥1 r£ηt(r)≤sup r≥1 r£η0(r)Ψ(t)+Πρ2, where Ψ(t) = 2(t+ 2)e−δt and Π = 2 Z∞ 0 sµ(s)ds + 2 Z∞ 1 s3µ(s)ds, which is finite. Obviously, if we only require that ku(t)k ≤ ρfor every t∈[τ, T ], then the results above are true on [τ, T]. Therefore, a straightforward consequence of Lemmas 4 and 5 is: Corollary 6. If η0∈D(T)and u∈L∞(τ, T;V)for every T > τ. Then ηt∈D(T)for all t>τ. Lemma 7. ([7, Lemma 3.7]) Let ϕbe a nonnegative absolutely continuous function on [0,∞) satisfying, for some ν > 0and 0≤σ < 1, the differential inequality d dtϕ+νϕ ≤g(1 + ϕσ), where gis a nonnegative function fulling sup t≥0Zt+1 t g(y)dy < ∞. Then there exists a constant C=C(σ, ν, g)such that, ϕ(t)≤1 1−σϕ(0)e−νt +C, ∀t≥0. NONLOCAL PDE WITH LONG TIME MEMORY 9 3. Existence of solutions to nonlocal differential equations with memory. As we did in the previous sections, in order to set up equation (4) in the framework of dynamical systems, we actually study transformed equation (9) by Dafermos. One natural question might be whether this procedure is consistent, namely, whether there is a link between equations (4) and (9). Indeed, it turns out that they are equivalent, or, to be more precise, the transformed equation (9) is in fact a generalization of the original equation (4), for more explanation, see [11]. Especially, when η0∈D(T), problems (4) and (9) are the same. By denoting z(t) = (u(t), ηt),and z0= (u0, η0), setting Lz=a(l(u))∆u+Z∞ 0 µ(s)∆η(s)ds, u +Tη, and G(z)=(−f(u) + g, 0). Then problem (9) can be written in the following compact form,      zt=Lz+G(z), z(x, t) = 0, z(x, τ) = z0, in Ω ×(τ, ∞), on ∂Ω×(τ, ∞), in Ω. (14) We are now ready to state the main result in this section. Theorem 8. Suppose (5),(11) and (h1)-(h2)hold true, let g∈H. In addition, assume that a(·) is locally Lipschitz, and there exists a positive constant ˜msuch that, a(s)≤˜m, ∀s∈R.(15) Then: (i)For any z0∈ H, there exists a unique solution z= (u, η)to problem (14) which satisfies u∈L∞(τ, T;H)∩L2(τ, T ;V)∩L2p(τ, T ;L2p(Ω)),∀T > τ, η∈L∞(τ, T;L2 µ(R+;V)),∀T > τ. Furthermore, z∈C(τ, T;H)for every T > τ, and the mapping F:z0∈ H → z(t)∈ H is continuous for every t∈[τ, T ]; (ii)For any z0∈ V, the unique solution z= (u, η)to problem (14) satisfies u∈L∞(τ, T;V)∩L2(τ, T ;D(A)),∀T > τ, η∈L∞(τ, T;L2 µ(R+;D(A))),∀T > τ. In addtion, z∈C(τ, T;V)for every T > τ. Proof. (i) The proof is divided into 5 steps. Step 1. (Faedo-Galerkin method) Recall that there exists a smooth orthonormal basis {wj}∞ j=1 in Hwhich is also orthogonal in V. In fact, it is a complete set of normalized eigenfunctions for −∆ in Vsuch that −∆wj=λjwj, where λjis the eigenvalue corresponding to wj. Meanwhile, we choose an orthonormal basis {ζj}∞ j=1 of L2 µ(R+;V) which also belongs to D(R+;V). 16 NONLOCAL PDE WITH LONG TIME MEMORY 4.1. Existence of absorbing sets in Hand V.We prove the existence of absorbing sets in both phase spaces, Hand V, in this subsection. Lemma 10. Under assumptions of Theorem 8, it holds: (i)If the initial value z0∈ H, then the solution to problem (14) in the weak sense satisfies, kz(t)k2 H≤ kz0k2 He−c1t+Λ c1 ,∀t≥0, where Λis given in Theorem 8 and c1= min{mλ1, δ}; (ii)If the initial value z0∈ V and c1> d0(cf. (36)), then kz(t)k2 V≤ kz0k2 Ve−(c1−d0)t+2 m(c1−d0)f2 2p−1|Ω|+|g|2,∀t≥0. Proof. We begin by proving (i). By (23) and (h1)-(h2), we obtain 2(((ηt)0, ηt))µ=−Z∞ 0 µ0(s)|∇ηt(s)|2ds ≥δZ∞ 0 µ(s)|∇ηt(s)|2ds =δkηtk2 µ. Thus, from (22) and the above estimation, together with the Poincar´e inequality, we arrive at d dtkzk2 H+mλ1|u|2+δkηtk2 µ+f0|u|2p 2p≤2a0|Ω|+1 mλ1 |g|2= Λ. Since c1= min{mλ1, δ}, we have d dtkzk2 H+c1kzk2 H≤Λ. By the Gronwall lemma, we obtain the desired result. (ii) In order to achieve uniform estimation involving the existence of a bounded absorbing set in V, let us proceed like in the previous proof. Integration by parts and (h1)-(h3) yield, 2(((ηt,(ηt)0)))µ=−Z∞ 0 µ0(s)|∆ηt(s)|2ds ≥δZ∞ 0 µ(s)|∆ηt(s)|2ds =δ|||ηt|||2 µ. Therefore, by means of the Poincar´e inequality, (37) can be written as d dtkzk2 V+mλ1kuk2+δ|||ηt|||2 µ≤2 mf2 2p−1|Ω|+d0kuk2+2 m|g|2. Since c1> d0, the above inequality becomes d dtkzk2 V+ (c1−d0)kzk2 V≤2 mf2 2p−1|Ω|+2 m|g|2. The proof of assertion (ii) is complete thanks to the Gronwall lemma.  4.2. Asymptotic compactness. Throughout this subsection, we will focus on the asymptotic compactness of the semigroup S(t) in phase space H. Let us start with the following lemma which is an immediate consequence of Lemma 10(i). Theorem 11. Let assumptions of Theorem 8 hold true. Then, there exists a positive constant RH, such that the ball BH:= B(0, RH)is an absorbing set for S(t)on H. Namely, for any given bounded set B:= B(0, R)⊂ H, there exists tH=tH(B)such that, kS(t)z0k2 H≤Re−c1t+Λ c1 ,∀t≥tH. NONLOCAL PDE WITH LONG TIME MEMORY 17 Our main purpose is to prove for any zn 0∈B⊂ H, for any sequence tn→ ∞ as n→ ∞, zn(tn) := z(tn; 0, zn 0) = S(tn)zn 0is relatively compact in H. To this end, by the well-known result that the embedding V ,→His compact, together with the fact in Lemma 3 that Eis relatively compact in L2 µ(R+;V), it is enough to prove the sequence zn(tn) is bounded in V× E. Theorem 12. Suppose the conditions of Theorem 8 are true. Then there exists a positive constant RE>0, such that the set BE:= B(0, RE)is an absorbing set for S(t)on V× E. Proof. The proof of this theorem is based on the subsequent lemma.  Lemma 13. Under assumptions of Theorem 8. Then, for any given constants RE 0>0and RH 0>0, there exist a positive constant Cand a positive function Ψ1vanishing at infinity, such that for any initial value z0= (u0, η0)satisfying kz0k2 V×E ≤RE 0and kz0k2 H≤RH 0, there holds: kS(t)z0k2 V×E ≤RE 0Ψ1+C, ∀t≥0. Proof. Thanks to Lemma 10(ii), we have kS(t)z0k2 V≤RE 0e−(c1−d0)t+2 m(c1−d0)f2 2p−1|Ω|+|g|2,∀t≥0,(39) here we used the fact that kz0k2 V≤ kz0k2 V×E ≤RE 0. It remains to show the required control on the last two terms of the norm of ηt(s) on E. From (39), we know there exsits t1=t(RE 0)>0, such that ku(t)k2≤c:= 1 + 2 m(c1−d0)f2 2p−1|Ω|+|g|2, for all t≥t1>0. Thus, by lemmas 4 and 5, we have kTηtk2 L2 µ(R+;V)+ sup x≥1 x£ηt(x)≤(RE 0e−δt + Ψ(t)+c, ∀t≥t1, cRE 0,∀t∈[0, t1].(40) With the help of Corollary 6, we finish the proof of this lemma.  In what follows, we need to prove for any bounded set B⊂ H, distH(S(t)B, BE)→0 holds as t→ ∞. Theorem 14. Suppose the conditions of Theorem 8 hold. Then, there exists an increasing positive function Φ, such that, up to (possibly) enlarging the radius RE, distH(S(t)B, BE)≤Φ(R)e−kt,∀t≥0, for every bounded set B:= B(0, R)⊂ H, where k= min{2mλ1, δ}. In view of Theorem 11, it is enough to show that distH(S(t)BH, BE)≤RHe−kt,∀t≥0. The proof of this fact is based on a suitable decomposition of the solutions S(t)z0. Recalling assumption (11) on f, noticing Remark 1(iii), we set f0(u) = f(u) + k4u, ∀u∈R. Obviously, f0 0(u)≥0 for every u∈R, then, for initial value z0= (u0, η0) of problem (4), we write S(t)z0as the sum S(t)z0=L(t)z0+K(t)z0, 18 NONLOCAL PDE WITH LONG TIME MEMORY where L(t)z0= (v(t), ξt) and K(t)z0= (w(t), ζt) solves the following problems,          vt−a(l(u))∆v−Z∞ 0 µ(s)∆ξt(s)ds +f0(u)−f0(w)=0, ξt t=−ξt s+v, (v(0), ξ0)=(u0, η0), in Ω ×(0,∞), in Ω ×(0,∞)×R+, in Ω, (41) and          wt−a(l(u))∆w−Z∞ 0 µ(s)∆ζt(s)ds +f0(w)−k4u=g, ζt t=−ζt s+w, (w(0), ζ0) = (0,0), in Ω ×(0,∞), in Ω ×(0,∞)×R+, in Ω. (42) Remark 15. Notice that, in general, L(t)and K(t)are not semigroups. The following lemmas furnish some properties of these two mappings, Land K, respectively. Lemma 16. Under assumptions of Theorem 8, let L(t)z0satisfy equation (41). Then, there exists a constant k= min{2mλ1, δ}, such that, sup z0∈B kL(t)z0k2 H≤RHe−kt,∀t≥0. Proof. Multiplying the first and second equations of problem (41) by vand ξtin spaces H and L2 µ(R+;V), respectively. Taking into account (5) and (h2), adding the results and using the Poincar´e inequality, it yields, d dt |v|2+kξtk2 µ+ 2mλ1|v|2+δkξtk2 µ+ 2(f0(u)−f0(w), u −w) = 0. Since f0 0(u)≥0, for all u∈R, we have d dtkL(t)z0k2 H+kkL(t)z0k2 H≤0. By the Gronwall lemma, we finish the proof of this lemma.  Lemma 17. Under assumptions of Theorem 8, let K(t)z0satisfy equation (42). Then, there holds, sup t≥0 sup z0∈B kK(t)z0k2 V×E ≤C, for some constant C=C(RH). Proof. We proceed similarly as in the proof of Lemma 16. Multiplying the first and second equations of problem (42) by −∆wand ζtin spaces Vand L2 µ(R+;D(A)), respectively. Taking into account (5) and (h2) and adding the results, we have, d dt kwk2+k|ζtk|2 µ+ 2m|∆w|2+δ|||ζt|||2 µ+ 2(∇(f0(w)−k4u),∇w) = 2(∇g, ∇w).(43) Since f0 0(w)≥0 for all w∈R, integration by parts implies, 2(∇f0(w),∇w) = 2(f0 0(w)∇w, ∇w)≥0. By the Young inequality, we derive 2(∇g, ∇w)≤2 m|g|2+m 2|∆w|2,2k4(u, ∆w)≤m 2|∆w|2+2 mk2 4|u|2. NONLOCAL PDE WITH LONG TIME MEMORY 19 Therefore, neglecting the fourth term of left hand side of (43), making use of the above estimations and the Poincar´e inequality, (43) can be written as, d dtkK(t)z0k2 V+kkK(t)z0k2 V≤2 m(k2 4|u|2+|g|2). Because u(t) is uniformly bounded on H(cf. Lemma 16), together with the initial value (w(τ), ζτ) = (0,0), by using the uniform Gronwall lemma (see, Lemma 7), we have sup t≥0 sup z0∈B kK(t)z0k2 V≤C. The proof of this lemma is finished by applying Lemmas 4 and 5, combining with Corollary 6.  Proof of Theorem 14. By Lemmas 16 and 17, we see at once the desired inequality, distH(S(t)B, BE) = sup z0∈B inf y∈BE kL(t)z0+K(t)z0−ykH ≤sup z0∈B kL(t)z0kH+ distH(K(t)B, BE) ≤RHe−kt. The result holds true since we choose REbigger than C(RH).  Theorem 18. Assume the conditions of Theorem 8 hold true. Then the semigroup S(t)defined in H×L2 µ(R+;V), i.e., H, associated to problem (14) has a global attractor A⊂H. 5. Conclusions and further comments. We have analyzed the dynamics of a nonlocal partial differential equation with long time memory (4) by performing a transformation, first used by Dafermos, which allows us to rewrite the original equation (4) with memory as a system (14) without delay. Next, all the results related to problem (14) are shown assuming the initial values (u0, η0)∈H×L2 µ(R+;V). 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