Non-autonomous nonlocal partial differential equations with delay and memory

Non-autonomous nonlocal partial differential equations with delay and memory Jiaohui Xu, Zhengce Zhang School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, PR China Tom´ as Caraballo∗ Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain Abstract The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space Cρ(X). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in L2(Ω), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, Eγ, has to be constructed. Keywords: Non-autonomous nonlocal parabolic equations; Bounded and unbounded delays; Stationary solutions; Pullback attractors. 1 Introduction It is well-known reaction-diffusion equations have been frequently used to model a great amount of phenomena in the real world. However, in recent decades, nonlocal (reaction-diffusion) problems have been investigated with great interest due to its usefulness in real applications (e.g. [1, 3, 16, 20, 26]). The nonlocal character in the equations has different sources. For example, the model in [26] focused on the role of nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. The authors in [4] studied nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions. ∗Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (Z. Zhang), [email protected] (T. Caraballo.) 1 One typical nonlocal problem, motivated by the mathematical modeling of a variety of phenomena coming from industrial applications ([27]), and shear banding formation in high strain metals ([3]), etc., has been studied by N. I. Kavallaris et al. [20]. The authors considered the following nonlocal stochastic parabolic problem          ∂u ∂t = ∆u+F(u) + σ(u)∂tW(x, t), u(x, t) = 0, u(x, t) = ξ(x), (x, t)∈DT:= D×(0, T), (x, t)∈∂D ×(0, T ), x∈D, where T > 0 and Dis a bounded subset of Rd,d≥1, with smooth boundary. Here the nonlocal term F(u) is defined by F(u) := λeu RDf(u)dxq, q > 0, for some positive constant λ. The authors examined in [17] one model of single-species dynamics which incorporates nonlocal effects, comparing with the standard approach to model a single species domain Ω of “Kolmogorov” type, ut= ∆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 [13] 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. [11, 12, 13, 14]), where l∈ L(L2(Ω); R), for instance, if g∈L2(Ω), l(u) = lg(u) = ZΩ g(x)u(x)dx. Then, the following nonlocal reaction-diffusion equation has received much attention due to its wide application (such as, in the ecological context, the interactions in single-species population dynamics cannot be local), ∂u ∂t −a(l(u))∆u=f(u),(1.1) where the function a∈C(R;R+), and it should fulfil natural conditions of non-degeneracy to avoid the extinction and the existence of solutions only in finite time intervals. Therefore, assume there exist positive constants m,M > 0 such that 0< m ≤a(s)≤M, ∀s∈R.(1.2) 2 Amongst the many notable results regarding such kind of nonlocal equations, it is worth stressing the importance of the work in [10]. In this paper, the authors studied a non-autonomous nonlocal parabolic equation similar to (1.1) but with an extra time-dependent term h(t), which allows to model more complex situations. On the other hand, in real life, time delays can arise everywhere, since every process, whether it is long or short, would consume time. As a motivation, we can mention a simple situation concerning the hemodynamical behavior of a person who has a decompensation of the glucose level (either high or low). Then, this person can be prescribed some intravenous insulin to compensate the level. Since the introduction of the drug in the bloodstream needs time to take effect, the above situation can be interpreted as a delay problem. Therefore, there is an increasing interest in studying biological, physical models with delay, for example, see [2, 6, 7, 8, 9, 21, 22, 25, 28] and the references therein. Consequently, in this context it is natural to take into account the effect of delay in the investigation of non-autonomous nonlocal partial differential equations, which leads us to study the following problem:          ∂u ∂t −a(l(u))∆u=f(u) + h(t, ut) u= 0 uτ(x, t) = ϕ(x, t) in Ω ×[τ, ∞), on ∂Ω×[τ, ∞), in Ω ×(−ρ, 0], (1.3) where Ω ⊂RNis a bounded open set, τ∈R, function a∈C(R;R+) is locally Lipschitz fulfilling (1.2), f∈C(R) and hcontains hereditary characteristics involving delays. Moreover, 0 < ρ ≤ ∞, which implies, we consider both cases, bounded and unbounded delays in this manuscript. From the technical point of view, some new difficulties appear with respect to all these works. Due to the peculiarities of delay problems, we have to work in a phase space Cρ(X) or Eγ which require a much more involved analysis to prove asymptotic compactness. Thanks to the Ascoli-Arzel`a theorem and energy estimations, we could prove long-time behavior successfully. It is also worth observing that variable delay, distributed delay and long term-memory cases can be contained in our functional delay term h(t, ut). In fact, the authors in [9] worked on the following reaction-diffusion equation with memory: ∂u ∂t −∆u+Zt −∞ γ(t−s)∆u(x, s)ds +g(x, t, u(x, t)) = f1(x, t, u(x, t −h)), with Dirichlet boundary condition, and γpossesses the special form γ(t) = −γ0e−d0t. In this way the long-time behavior of solutions can be analyzed. However, if we consider a generalized kernel with singularity, γ(t) = e−d0t tα, where α∈(0,1), the long-time behavior of this problem cannot be handled with our analysis. One of our main purposes is to prove existence, uniqueness and regularity properties of weak solutions to problem (1.3). To this end, the well-known Galerkin method is adapted to our 3 problem properly. Moreover, we are able to analyze some stability properties (cf. Definition 3.3) of stationary solutions to problem (1.3), when the delay term has special forms (variable delay/ distributed delay). By a direct approach or constructing Lyapunov functionals, our goals are achieved successfully. At last, to make our analysis more complete, we establish the existence of pullback attractors in L2(Ω) in the bounded and unbounded delay cases. We want to emphasize that, due to the lack of asymptotic compactness for the unbounded delay case, it is impossible to study the existence of pullback attractors in L2(Ω) if the phase space is defined by C∞(L2(Ω)). Instead, the existence of pullback attractor is proved considering the phase space Eγ. This paper is structured as follows. Section 2 is devoted to the existence, uniqueness and continuous dependence on initial values of weak solutions to (1.3) by the Galerkin method. The existence of strong solutions is also shown by energy estimations. In Section 3, existence and uniqueness of stationary solutions are established and several methods are used to analyze the stability of the stationary solutions. We consider the existence of pullback attractors in the last section, providing more details about their structure in the bounded delay case. 2 Well-posedness to nonlocal partial differential equations with delay Here we consider the following nonlocal parabolic equation with bounded or unbounded delay,    ∂u ∂t −a(l(u))∆u=f(u) + h(t, ut) u= 0 in Ω ×[τ, ∞), on ∂Ω×[τ, ∞),(2.1) with initial value u(x, τ +θ) = ϕ(x, θ), θ ∈(−ρ, 0], x ∈Ω, where Ω ⊂RNis a bounded open set and the initial time τ∈R. The function a∈C(R;R+) and there exists a constant msuch that 0< m ≤a(s),∀s∈R.(2.2) In addition, we assume that l∈ L(L2(Ω),R), f∈C(R) and there exist constants η > 0, Cf>0 such that |f(s)| ≤ Cf(1 + |s|),∀s∈R,(2.3) (f(s)−f(r))(s−r)≤η(s−r)2,∀s, r ∈R.(2.4) Consider a fixed T > 0, given a function u: (τ−ρ, T]→L2(Ω), for each t∈(τ, T ], we denote by utthe function defined on (−ρ, 0] via the relation ut(s) = u(t+s), s ∈(−ρ, 0], where 0 < ρ ≤ ∞. 4 In order to state the problem in the correct framework, let us first establish some suitable assumptions on the term where the delay is present. In a general way, let Xand Ybe two separable Banach spaces, and Cρ(X) = {φ∈C((−ρ, 0]; X); lim s→−ρφ(s) exists in X}, which is a Banach space with the standard sup norm kαkCρ(X)= sup−ρ≤θ≤0kα(θ)kX, and let h:R×Cρ(X)→Y, satisfy (H1) for all ξ∈Cρ(X), the mapping t∈(τ, T ]→h(t, ξ)∈Xis measurable; (H2) For each t∈(τ, T ], h(t, 0) = 0; (H3) There exists Lh>0 such that for all t∈(τ, T], α, β ∈Cρ(X), kh(t, α)−h(t, β)kY≤Lhkα−βkCρ(X); (H4) There exists Ch>0 such that for all t∈(τ, T], u, v ∈C((τ−ρ, T]; X), Zt τkh(s, us)−h(s, vs)k2 Yds ≤ChZt τ−ρku(s)−v(s)k2 Xds. Remark 2.1 (i) In order to simplify notation, we denote Cρ(X) := C((−ρ, 0]; X), which represents: when it is bounded delay, Cρ(X) := C([−ρ, 0]; X); when it is unbounded delay, Cρ(X) := C((−∞,0]; X). (ii) Notice that when we consider bounded delay, namely ρ > 0is a constant, then the space Cρ(X) := {φ∈C([−ρ, 0]; X)}, because if φis a continuous function defined on [−ρ, 0], lims→−ρφ(s)exists automatically. The additional condition only makes sense for unbounded delay. (iii) Indeed, for the unbounded delay case ρ=∞, we have C∞(X) := C((−∞,0]; X) = {φ∈C((−∞,0]; X); lim s→−∞ φ(s)exists in X}, which is a Banach space equipped with the norm kφkC∞(X)= sup θ∈(−∞,0] kφ(θ)kX. Throughout this paper, we make use of several notations which are introduced in what follows. Let Ω be an open bounded set of RN, (·,·) and | ·| denote the L2-inner product and L2-norm, ((·,·)) and k·k denote the H1 0-inner product and H1 0-norm. Recall that for every v∈H1 0(Ω), the Poincar´e inequality λ1(Ω)|v|2≤ kvk2,(2.5) 5 holds, where λ1(Ω) is related to the domain Ω and is the first eigenvalue of −∆ with the zero Dirichlet boundary conditions. In the sequel, unless otherwise specified, we write λ1instead of λ1(Ω) for simplicity. From now on, we identify L2(Ω) with its dual. Therefore, we have the usual chain of dense and compact embeddings H1 0(Ω) ⊂L2(Ω) ⊂H−1(Ω). Thanks to the previous identification, l(u) is in fact (l, u). However, we keep the usual notation in the existing previous literature l(u) instead of (l, u) for the operator lacting on u. Now we will show the definitions of solutions to problem (2.1). Definition 2.2 A weak solution to (2.1), corresponding to the initial value ϕ∈Cρ(L2(Ω)), is a function u∈L2((τ, T]; H1 0(Ω)) ∩L∞((τ, T]; L2(Ω)) for all T > τ, such that u(t) = ϕ(t−τ)for t∈(τ−ρ, τ], and d dt(u(t), v) + a(l(u(t)))((u(t), v)) = (f(u(t)), v)+ < h(t, ut), v >, ∀v∈H1 0(Ω),(2.6) where the previous equation must be understood in the sense of D0(τ,∞). A notion of more regular solutions is also suitable. Definition 2.3 A strong solution to (2.1) is a weak solution uwhich also satisfies that u∈ L2((τ, T]; D(−∆)) ∩L∞((τ, T]; H1 0(Ω)) for all T > τ. To prove the existence of weak and strong solutions to problem (2.1), we will use the Faedo- Galerkin approximations and pass to the limit by using compactness arguments. Thanks to spectral theory, there exists a sequence {wi}i≥1, which is a Hilbert basis of L2(Ω) consisting of the eigenfunctions of −∆ in H1 0(Ω). Firstly, we consider the function un(t) := Pn j=1 γnj(t)wj, for all n≥1,    d dt (un(t), wj) + a(l(un(t)))((un(t), wj)) = (f(un(t)), wj)+ < h(t, un t), wj>, (un(τ+θ), wj) = (ϕ(θ), wj) t∈(τ, ∞), θ∈(−ρ, 0], (2.7) where j= 1,2,··· , n. Observe that (2.7) is a Cauchy problem to the following ordinary differential system in Rn, dγnj(t) dt +λja(l(un(t)))γnj(t)=(f(un(t)), wj)+ < h(t, un t), wj>, j = 1,2,··· , n, (2.8) where t≥τ,λjis the eigenvalue associated to the eigenfunction wjand the vector (γn1,··· , γnn) is unknown. Proposition 2.4 Suppose that a∈C(R;R+)fulfils (2.2), f(·)∈C(R)verifies (2.3), l∈L2(Ω), h(·,·)satisfies conditions (H1)-(H4), with X=H1 0(Ω) and Y=H−1(Ω), for an initial time τ∈R. 6 Then for each initial value ϕ∈Cρ(L2(Ω)), there exists at least one local solution (γn1,··· , γnn) of the ordinary differential system (2.8) defined on some interval (τ, tn]. Furthermore, if the function ais locally Lipschitz, fsatisfies (2.4), and m2≥4Ch,(2.9) the uniqueness of local solution to problem (2.8) is guaranteed. Proof. We split the proof into several steps. Step 1: Existence of local solution. To do this, we are going to use the method of [18, Section 2.6], which is a generalization of Peano’s Theorem to delay problem. Define g: Θ →Rn, (t, ξ)7−→ (−λ1z(ξ(0))ξ1(0) + (f( n X i=1 ξi(0)wi), w1)+ < h(t, n X i=1 ξiwi), w1>+··· , −λnz(ξ(0))ξn(0) + (f( n X i=1 ξi(0)wi), wn)+ < h(t, n X i=1 ξiwi), wn>), where Θ = {(t, ξ)∈(τ, T]×Cρ(Rn) : |ξ−((ϕ, w1),··· ,(ϕ, wn))| ≤ b}, for any fixed b∈R+and ξ= (ξ1,··· , ξn)7−→ z(ξ(0)) = a(l( n X i=1 ξi(0)wi)).(2.10) Firstly, we are going to prove that gis a Caratheodory function. Consider ξ∈Cρ(Rn) fixed, the function g(·, ξ) is measurable, because gj(·, ξ) = −λjz(ξ(0))ξj(0) + f(( n X i=1 ξi(0)wi), wj)+ < h(t, n X i=1 ξiwi), wj> is a measurable function as a consequence of Fubini’s Theorem and condition (H1). Secondly, we need to check that the function g(t, ·) is continuous a.e. t∈(τ, T]. Indeed, gj(t, ξ) = −λjz(ξ(0))ξj(0) + (f( n X i=1 ξi(0)wi), wj)+ < h(t, n X i=1 ξiwi), wj> is a continuous function with respect to the second variable, because the functions z(ξ(0)), ξ(0) ∈ Rn→(f(Pn i=1 ξi(0)wi), wj), and ξ∈Cρ(Rn)→< h(t, Pn i=1 ξiwi), wj>are continuous. Thirdly, we are about to prove for each (t, ξ)∈Θ, there exists a Lebesgue measurable function m(·)∈L1(τ, T) such that |g(¯ t, ¯ ξ)| ≤ m(t),∀(¯ t, ¯ ξ)∈V(t, ξ), 7 where V(t, ξ)⊂Θ is a neighborhood of (t, ξ). From the definition of Θ, there exists a constant CΘ(depending on ϕ) such that |ξ| ≤ b+|((ϕ, w1),··· ,(ϕ, wn))| ≤ CΘ. Next (2.3) implies that (f( n X i=1 ξi(0)wi), wj)≤21 2Cf|Ω|1 2|wj|+ 21 2CfCΘ( n X i=1 |ξi(0)|)|wj|. At last, applying conditions (H2)-(H3), we obtain < h(t, n X i=1 ξiwi), wj>=<(h(t, n X i=1 ξiwi)−h(t, 0)), wj> ≤ kh(t, n X i=1 ξiwi)−h(t, 0)kH−1(Ω))kwjk ≤Lhk n X i=1 ξiwikCρ(H1 0(Ω))kwjk ≤LhkξkCρ(Rn) n X i=1 kwik2!1/2 kwjk ≤LhCΘ n X i=1 kwik2!1/2 kwjk. Taking into account (2.10) and owing to the continuity of the function ain the compact interval I0:= [−|l|CΘPn i=1 |wi|,|l|CΘPn i=1 |wi|], there exists a constant M > 0 such that z(ξ)≤M, ∀ξ∈Cρ(Rn),|ξ| ≤ CΘ. Bearing this in mind, we obtain |gj(t, ξ)| ≤ λjz(ξ(0))|ξj(0)|+|(f( n X i=1 ξi(0)wi), wj)|+|< h(t, n X i=1 ξiwi), wj>| ≤λjMCΘ+ 21 2Cf|Ω|1 2|wj|+ 21 2CfCΘ( n X i=1 |ξi(0)|)|wj| +LhCΘ n X i=1 kwik2!1/2 kwjk ∈ L1(τ, T). In conclusion, there exists a local solution to problem (2.8). Step 2: Uniqueness of local solution. Since the function ais locally Lipschitz, for any bounded interval [−R, R] of R, there exists a positive constant La(R) such that |a(x)−a(y)| ≤ La(R)|x−y|,∀x, y ∈[−R, R].(2.11) 8 Assume that there exist two solutions $n(t), ϑn(t) of the ordinary differential system (2.8) in (τ, t1] and (τ, t2] respectively. Then it holds    ($nj(t)−ϑnj(t))0=gj(t, $n t)−gj(t, ϑn t), ($nj −ϑnj)(t) = 0, t∈(τ, min{t1, t2}], t∈(τ−ρ, τ],(2.12) where gj(t, $n t) = −λj(a(l(un(t))))$nj(t)+(f(un(t)), wj)+ < h(t, un t), wj>, gj(t, ϑn t) = −λj(a(l(vn(t))))ϑnj(t)+(f(vn(t)), wj)+ < h(t, vn t), wj>, un(t) := Pn j=1 $nj(t)wjand vn(t) := Pn j=1 ϑnj(t)wj. Multiplying (2.12) by $nj −ϑnj, summing from j= 1 to nand making use of (2.4), we obtain, 1 2 d dt|un(t)−vn(t)|2+a(l(un(t)))kun(t)−vn(t)k2 ≤ |a(l(un(t))) −a(l(vn(t)))||((un(t), un(t)−vn(t)))|+η|un(t)−vn(t)|2 +<(h(t, un t)−h(t, vn t)), un(t)−vn(t)> . It is straightforward to deduce from (2.7) that un,vn∈C((τ, min{t1, t2}]; L2(Ω)). When ρ > 0 is a constant, obviously, there exist a constant M0>0 and some R > 0, such that |un(t)|2≤M0and |vn(t)|2≤M0,∀t∈(τ, min{t1, t2}].(2.13) When ρ=∞, by definition of the phase space C∞(L2(Ω)), the limit of un(t) and vn(t) exist as tgoes to minus infinity, (2.13) also holds. In addition, taking into account that l∈L2(Ω), it satisfies {l(un(t))}t∈(τ,min{t1,t2}]∈[−R, R], {l(vn(t))}t∈(τ,min{t1,t2}]∈[−R, R]. Hence, using (2.2), (2.4), (H4) and the fact that function ais locally Lipschitz (with Lipschitz constant La(·), cf. (2.11)), integrating the above inequality between τand t, we deduce that |un(t)−vn(t)|2+m 2Zt τkun(s)−vn(s)k2ds ≤ |un(τ)−vn(τ)|2+(La(R))2|l|2λn(M0)2 m+ 2ηZt τ|un(s)−vn(s)|2ds +2Ch mZt τkun(s)−vn(s)k2ds. With the help of (2.9), we have |un(t)−vn(t)|2≤ |un(τ)−vn(τ)|2eCt, where C=(La(R))2|l|2λn(M0)2+ 2mη m. The proof of this proposition is finished.  9 From this and taking into account the boundedness of {un}in L2((τ, T ]; H1 0(Ω)), we deduce that {un}is bounded in L∞((τ, T ]; H1 0(Ω)). As a byproduct, the boundedness of {un} in L2((τ+ε, T]; D(−∆)) holds true just taking s=τand t=Tin (2.22). In addition, by means of this more regular boundedness, we deduce that the sequence {(un)0}is bounded in L2((ε+τ, T]; L2(Ω)). Thanks to the uniqueness of the weak solution, {un}converges to uweakly in L2((τ+ε, T]; D(−∆)) and {(un)0}converges to {u0}weakly in L2((τ+ε, T]; L2(Ω)). As a consequence, u∈L2((τ+ε, T]; D(−∆)) ∩C((τ, T]; H1 0(Ω)). Step 2: Strong solution. In this step, if ϕ∈L2((−ρ, 0]; H1 0(Ω)), we will show that u∈ L2((τ, T]; D(−∆))∩C((τ, T]; H1 0(Ω)) for all T > τ. To that end, we multiply (2.7) by λjγnj, sum from j= 1 to nand use (2.2), obtaining 1 2 d dtkun(t)k2+m|−∆un(t)|2≤(f(un(t)),−∆un(t))+(h(t, un t),−∆un(t)),a.e. t∈(τ, T].(2.23) Now, from (2.3) and the Young inequality, we have (f(un(t)),−∆un(t)) ≤1 m|f(un(t))|2+m 4|−∆un(t)|2 ≤2C2 f|Ω| m+2C2 f m|un(t)|2+m 4|−∆un(t)|2, and by (H2), (h(t, un t),−∆un(t)) ≤1 m|h(t, un t)−h(t, 0)|2+m 4|−∆un(t)|2. Taking this into account, from (2.23), we deduce that d dtkun(t)k2+m|−∆un(t)|2≤4C2 f|Ω| m+4C2 f mλ1kun(t)k2+2 m|h(t, un t)−h(t, 0)|2,a.e. t > τ. (2.24) Integrating (2.24) between τand t,t∈(τ, T], by (H4) we have kun(t)k2+mZt τ|−∆un(s)|2ds ≤ kϕ(0)k2+4C2 f|Ω|(T−τ) m+ 4C2 f+ 2Ch mλ1!Zt τkun(s)k2ds +2Ch mλ1kϕk2 L2((−ρ,0];H1 0(Ω)). Since {un}is bounded in L2((τ, T]; H1 0(Ω)), the sequence {un}is bounded in L∞((τ, T]; H1 0(Ω)) ∩L2((τ, T]; D(−∆)). As a result, the sequence {(un)0}is bounded in L2((τ, T]; L2(Ω)). Thanks to the uniqueness of weak solution, {un}converges to uweakly-star in L∞((τ, T ]; H1 0(Ω)) and weakly in L2((τ, T]; D(−∆)), also {(un)0}converges to u0weakly in L2((τ, T]; L2(Ω)). Therefore, we have u∈C((τ, T]; H1 0(Ω)), since u∈L∞((τ, T ]; H1 0(Ω))∩L2((τ, T]; D(−∆)), and u0∈L2((τ, T]; L2(Ω)) as well. The proof is finished.  16 3 Analysis of the stationary problem 3.1 Statement of the problem In this section, we are interested in studying the asymptotic behavior of the stationary (steady state) solutions of problem (2.1). Recall that a stationary (steady state) solution to our problem is an element u∗∈H1 0(Ω), such that −a(l(u∗))∆u∗=f(u∗) + h(t, u∗),∀t∈R. To state this problem, we need to assume h(t, u∗) is independent of t, i.e., there exists a mapping H:H1 0(Ω) →H−1(Ω) such that h(t, ξ) = H(ξ∗), whenever ξ(θ) = ξ∗, for all θ≤0. For instance, if H:R→Ris Lipschitz continuous and we define h(t, ξ) = H(ξ(−δ(t))) via the Nemytskii operator, for δ(·) : R+→(0, ρ), i.e., H(ξ(−δ(t)))(x) = H(ξ(−δ(t))(x)), then h(t, ξ∗) = H(ξ∗). Another example satisfying our problem is the case of distributed delay: h(t, ξ) = R0 −ρk(s, ξ(s))ds with k:R2→Ra Lipschitz continuous function with respect to its second variable. For more details, see [23]. This leads us to first analyze the following elliptic problem,    −a(l(u))∆u=f(u) + H(u), u= 0, in Ω, on ∂Ω,(3.1) where the functions aand fare globally Lipschitz, with respective Lipschitz constants La,Lf≥0 and there exist positive constants mand M, such that 0< m ≤a(s)≤M, ∀s∈R.(3.2) 3.2 The existence and uniqueness of stationary solutions Definition 3.1 A weak solution to (3.1) is an element u∗∈H1 0(Ω) such that a(l(u∗))((u∗, v)) = (f(u∗), v)+ < H(u∗), v >, ∀v∈H1 0(Ω). In what follows, we analyze the existence of solutions to (3.1) by a corollary of the Brouwer fixed point theorem. The uniqueness as well as the stability is also studied under suitable assumptions. Theorem 3.2 Let a,fand H:H1 0(Ω) →H−1(Ω) be globally Lipschitz functions, with Lipschitz constants La,Lfand LH, respectively. Assume (3.2) is also satisfied, l∈L2(Ω) and m > Lfλ−1 1+LH. Then: 17 1. There exists at least one solution to (3.1). In addition, any solution u∗to (3.1) fulfills ku∗k ≤ Υ := λ−1/2 1|Ω|1/2|f(0)|+kH(0)kH−1(Ω) m−λ−1 1Lf−LH .(3.3) 2. If the mapping H:L2(Ω) →L2(Ω), then every solution u∗to (3.1) belongs to D(−∆). 3. Under the additional assumption λ−1/2 1|l|2LaΥ< m −λ−1 1Lf−LH,(3.4) problem (3.1) possesses a unique solution. Proof. (i) Let us prove the first result. Consider an orthonormal Hilbert basis {wj:j≥1}of L2(Ω) consisting of the eigenvectors associated with eigenvalues {λj:j≥1}of the operator −∆ with zero Dirichlet boundary condition in Ω. For each n≥1, let us denote Vn= span[w1,··· , wn], with the inner product ((·,·)) and norm k·k. Now, we define operators Rn:Vn→Vn, for all n≥1, as follows: ((Rnu, v)) =<−a(l(u))∆u, v > −(f(u), v)−< H(u), v >, ∀u, v ∈Vn.(3.5) Observe that the right hand side of (3.5) is a continuous linear map from Vnto R, thanks to the Riesz Theorem, each Rnu∈Vnis well defined. In addition, by means of (3.2), the Poincar´e inequality and the Lipschitz continuity of the functions a,fand H, we deduce, ((Rnu−Rn˜u, v)) =<−a(l(u))∆u+a(l(˜u))∆˜u−f(u) + f(˜u)−H(u) + H(˜u), v > =<−a(l(u))∆(u−˜u)+(a(l(˜u)) −a(l(u))∆˜u, v > + (f(˜u)−f(u), v)+ < H(˜u)−H(u), v > ≤(M+La|l|λ−1/2 1k˜uk+Lfλ−1 1+LH)k˜u−ukkvk, for all u, ˜u,v∈Vn. Therefore, kRnu−Rn˜uk ≤ (M+La|l|λ−1/2 1k˜uk+Lfλ−1 1+LH)k˜u−uk, for all u, ˜u∈Vn. Namely, Rnis continuous. On the other hand, again by (3.2), the Poincar´e inequality and the global Lipschitz continuity of f, we have ((Rnu, u)) =<−a(l(u))∆u, u > −(f(u), u)−< H(u)−H(0), u > −< H(0), u > ≥mkuk2−Lfλ−1 1kuk2−|f(0)||Ω|1/2λ−1/2 1kuk−LHkuk2−kH(0)kH−1(Ω)kuk := (m−Lfλ−1 1−LH)kuk2−f(0)|Ω|1/2λ−1/2 1+kH(0)kH−1(Ω)kuk, 18 for all u∈Vn. Therefore, taking Υ := λ−1/2 1|Ω|1/2|f(0)|+kH(0)kH−1(Ω) m−λ−1 1Lf−LH , we obtain ((Rnu, u)) ≥0,∀u∈Vnsuch that kuk= Υ. As a consequence of a corollary of the Brouwer fixed point theorem ([22, Lemma 4.3]), we deduce that for each n≥1, there exists un∈Vnsuch that Rn(un) = 0, with kuk ≤ Υ.(3.6) Therefore, <−a(l(un))∆un, v >= (f(un), v)+ < H(un), v >, ∀v∈Vn.(3.7) Now, using the boundedness of {un}in H1 0(Ω) by Υ and the compact embedding H1 0(Ω) ,→ L2(Ω), we can extract a subsequence {unk}⊂{un}, such that    unk→u∗, unk→u∗, weakly in H1 0(Ω), strongly in L2(Ω),(3.8) where u∗∈H1 0(Ω) is a solution to (3.1). To check that, we just take limit in (3.7) and use assumptions imposed on a,l,fand H. Moreover, u∗fulfils (3.6). So far, we only have proved that there exists at least a solution u∗to problem (3.1) and verifies (3.3). But this does not imply that any solution u∗to problem (3.1) fulfils (3.3) since the uniqueness of solution is not guaranteed. Therefore, we are going to prove that any solution u∗ to problem (3.1) fulfils (3.3). Let u∗be a solution to (3.1). It holds mku∗k2≤ |f(u∗)−f(0)||u∗|+|f(0)||Ω|1/2|u∗| +< H(u∗)−H(0), u∗>+kH(0)kH−1(Ω)ku∗k. Since the functions fand Hare globally Lipschitz, by the Poincar´e inequality, we obtain mku∗k2≤(Lfλ−1 1+LH)ku∗k2+|f(0)||Ω|1/2λ−1/2 1+kH(0)kH−1(Ω)ku∗k. Therefore, each solution u∗of problem (3.1) satisfies (3.3). (ii) We will check that if H:L2(Ω) →L2(Ω), then every solution u∗to (3.1) belongs to D(−∆). In what follows, we represent (un)∗=Pnu∗:= Pn i=1(u∗, wi)wi. Since u∗is a solution to (3.1), taking v=−∆(un)∗in Definition 3.1, we deduce a(l(u∗))|−∆(un)∗|2= (f(u∗),−∆(un)∗)+(H(u∗),−∆(un)∗). 19 Using the Cauchy-Schwartz and Young inequalities, the fact that fand Hare globally Lipschitz and (3.3), we deduce (f(u∗),−∆(un)∗)≤1 m|f(u∗)|2+m 4|−∆(un)∗|2 ≤2 m|f(u∗)−f(0)|2+m 4|−∆(un)∗|2+2 mZΩ|f(0)|2dx ≤2L2 f mλ1ku∗k2+2 m|f(0)|2|Ω|+m 4|−∆(un)∗|2 ≤2λ−1 1L2 fΥ2+ 2|f(0)|2|Ω| m+m 4|−∆(un)∗|2, and (H(u∗),−∆(un)∗)≤1 m|H(u∗)|2+m 4|−∆(un)∗|2 ≤2L2 H mλ1ku∗k2+2 m|H(0)|2|Ω|+m 4|−∆(un)∗|2. Thus, |−∆(un)∗|2≤2 m22λ−1 1L2 fΥ2+ 2λ−1 1L2 HΥ2+ 2(|f(0)|2+|H(0)|2)|Ω|. Then, as the sequence {Pnu∗}is bounded in D(−∆) and Pnu∗converges to u∗strongly in L2(Ω), it holds that u∗∈D(−∆). (iii) Finally, we state the uniqueness to this problem. Assume u1and u2are two solutions of (3.1). Then <−a(l(u1))∆u1+a(l(u2))∆u2, v > = (f(u1)−f(u2), v)+ < H(u1)−H(u2), v >, ∀v∈H1 0(Ω). Adding ±a(l(u1))∆u2into the above equality, taking v=u1−u2, we obtain mku1−u2k2≤(λ−1/2 1|l|Laku2k+λ−1 1Lf+LH)ku1−u2k2. We now argue by contradiction, assume that u16=u2. Then, we can simplify the above expression dropping the factor ku1−u2k2. However, using a priori estimate (3.3) for u2, we would arrive at the opposite inequality to that one in (3.4), what is a contradiction. Therefore, u1=u2holds. The proof of this theorem is complete.  In the sequel, our goal is to establish sufficient conditions ensuring stability of stationary solutions to (2.1). We first recall the stability definitions we will use in our analysis later. Definition 3.3 (1) A stationary solution u∗to (2.1) is said to be stable, if for any ε > 0,τ∈R, there exists δ > 0such that if ϕ∈L2((−ρ, 0]; H1 0(Ω)) satisfies kϕ−u∗kL2((−ρ,0];H1 0(Ω)) < δ, then 20 the corresponding solution u(·;τ, ϕ)to (2.1) exists for all t≥τ, and satisfies |u(t;τ, ϕ)−u∗|< ε, for all t≥τ. (2) A stationary solution u∗to (2.1) is said to be attractive, if for every τ∈Rthere exists ˜ δ > 0such that for ϕ∈L2((−ρ, 0]; H1 0(Ω)) with kϕ−u∗kL2((−ρ,0];H1 0(Ω)) <˜ δ, the solution u(·;τ, ϕ) to (2.1) exists for all t≥τ, and satisfies limt→∞ |u(t;τ, ϕ)−u∗|= 0. (3) A stationary solution u∗to (2.1) is said to be asymptotically stable if it is stable and attractive. 3.3 Analysis of stability: a direct approach In this section, we prove the local stability of stationary solutions for a general delay term by doing energy estimations directly. Moreover, the asymptotic stability of stationary solutions to (2.1) with bounded delay will be presented later. Theorem 3.4 Assume the hypotheses of Theorem 3.2 and (H4). If u(·)is any solution to (2.1), and u∗is the unique stationary solution of the same problem, then |u(t)−u∗|2≤ |ϕ(0) −u∗|2+Ch mkϕ−u∗k2 L2((−ρ,0];H1 0(Ω)),∀t≥τ, provided 2La|l|ku∗k √λ1 +2Lf λ1 +Ch m≤m. (3.9) Proof. From the energy equality, 1 2 d dt|u(t)−u∗|2=< a(l(u(t)))∆u(t)−a(l(u∗))∆u∗+f(u) −f(u∗) + h(t, ut)−h(t, u∗), u(t)−u∗> . (3.10) Adding ±a(l(u))∆u∗in (3.10) and using (3.2), the Poincar´e and Young inequalities, the global Lipschitz continuity of the functions aand f, we have d dt|u(t)−u∗|2≤2a(l(u))(∆(u−u∗), u(t)−u∗) + 2(a(l(u))∆u∗−a(l(u∗))∆u∗, u(t)−u∗) + 2(f(u)−f(u∗), u(t)−u∗)+2< h(t, ut)−h(t, u∗), u(t)−u∗> ≤2−m 2+La|l|ku∗k √λ1 +Lf λ1ku(t)−u∗k2 +1 mkh(t, ut)−h(t, u∗)k2 H−1(Ω). Integrating the above expression between τand t, by (H4), we obtain |u(t)−u∗|2≤ |ϕ(0) −u∗|2+Ch mZ0 −ρkϕ−u∗k2ds + 2 −m 2+La|l|ku∗k √λ1 +Lf λ1 +Ch 2mZt τku(s)−u∗k2ds. 21 Neglecting the last term on the right-hand side of the above inequality since it is non-positive, it follows, |u(t)−u∗|2≤ |ϕ(0) −u∗|2+Ch mkϕ−u∗k2 L2((−ρ,0];H1 0(Ω)),∀t≥τ. The proof is complete.  Remark 3.5 Notice that, the sufficient condition (3.9) ensuring stability of the stationary solution involves its norm. It is possible to obtain another sufficient condition independently of ku∗k. Indeed, take into account that ku∗k≤kΥk, (see (3.3)). If we impose 2La|l|kΥk √λ1 +2Lf λ1 +Ch m≤m, then condition (3.9) holds automatically, which means this condition is stronger than (3.9). Next we will prove asymptotic stability of stationary solutions when the delay is bounded, in fact, we will prove exponential asymptotic stability. Theorem 3.6 Under the assumptions of Theorem 3.2, also, suppose there exist α0>0and C(ρ) (which may also depend on α0), such that for each α∈(0, α0], it holds Zt τ eαskh(s, us)−h(s, vs)k2 H−1(Ω)ds ≤C(ρ)Zt τ−ρ eαsku(s)−v(s)k2ds, (3.11) and m > 2La|l|ku∗k √λ1 +2Lf λ1 +C(ρ) m.(3.12) If problem (2.1) is driven by bounded delay, then for sufficiently small α, we have |u(t)−u∗|2≤e−α(t−τ)|ϕ(0) −u∗|2+C(ρ) mkϕ−u∗k2 L2((−ρ,0];H1 0(Ω)),(3.13) where u(·)is the solution to (2.1) corresponding to the initial value ϕ, and u∗is the unique stationary solution of (2.1). Proof. Let 0 < α ≤α0, then d dteαt|u(t)−u∗|2=αeαt|u(t)−u∗|2+eαt d dt|u(t)−u∗|2. Integrating the above equality between τand t, together with (3.10), eαt|u(t)−u∗|2≤ |ϕ(0) −u∗|2eατ +C(ρ) mZ0 −ρ eα(s+τ)kϕ(s)−u∗k2ds +α λ1Zt τ eαsku(s)−u∗k2ds +Zt τ eαs −m+2La|l|ku∗k √λ1 +2Lf λ1 +C(ρ) mku(s)−u∗k2ds. 22 Notice, if αis small enough, combined with assumption (3.12), the sum of last two terms of the above inequality keeps being negative. Therefore, |u(t)−u∗|2≤e−α(t−τ)|ϕ(0) −u∗|2+C(ρ) mkϕ−u∗k2 L2((−ρ,0];H1 0(Ω)). This finishes the proof.  Remark 3.7 The previous result (Theorem 3.6) is valid only for bounded delay, because condition (3.11) is not reasonable for unbounded delay, for more explanations, see [6]. 3.4 Asymptotic stability via the construction of Lyapunov functionals In this subsection, we will exploit the theory of Lyapunov functionals to complete our stability analysis in the case of unbounded variable delay. In general, using Lyapunov functionals (instead of Lyapunov functions) provides better stability results (cf. [19]), but the type of the delay terms must be more specific. In our case, we will restrict to the case of variable delay (cf. (3.15)), for simplicity, we will assume zero is the stationary solution. To this end, let us introduce an abstract problem, consider operators ˜ A(t, ·) : H1 0(Ω) →H−1(Ω) and ˜ f(t, ·) : C∞(H1 0(Ω)) →H1 0(Ω) with ˜ A(t, 0) = 0 and ˜ f(t, 0) = 0. Assume    du dt =˜ A(t, u) + ˜ f(t, ut), uτ(θ) = ϕ(θ), t∈[τ, ∞), θ∈(−∞,0],(3.14) is a well-posed problem, its solution, belonging to C∞(H1 0(Ω)), is defined globally in time. The stability of the trivial solution of (3.14) can be analyzed by constructing Lyapunov functionals. Moreover, if we improve the decay of the functional, we gain not only stability but asymptotic stability. Before proving the main theorem, let us recall the following well-known result (see, e.g. [23]). Proposition 3.8 Assume that there exist a functional V:R+×C∞(H1 0(Ω)) →R+and positive constants γ1,γ2such that, for any ϕ∈C∞(H1 0(Ω)), the solution u(·) = u(·;τ, ϕ)to (3.14) satisfies V(t, ut)≥γ1|u(t)|2, t ≥τ, V(τ, uτ)≤γ2kϕk2 C∞(H1 0(Ω)). Then: (i) If d dtV(t, ut)≤0for t≥τ, the trivial solution to (3.14) is stable; (ii) If there exists a positive constant γ3such that d dt V(t, ut)≤ −γ3|u(t)|2for t≥τ, the trivial solution to problem (3.14) is asymptotically stable. 23 We will apply the above results to the following equation, which is a particular case of (3.14), du dt =a(l(u))∆u+f(u) + H(u(t−ρ(t))),(3.15) where H:H1 0(Ω) →H1 0(Ω) is an appropriate function. Note that, assumption (3.2) implies (a(l(u))∆u, u)≤ −mkuk2, u ∈H1 0(Ω).(3.16) Additionally, based on the condition that functions Hare globally Lipschitz with Lipschitz constant LH, we have kH(u)k ≤ LHkuk, u ∈H1 0(Ω).(3.17) In the sequel, we will present the main result of this subsection. Theorem 3.9 Assume that operators in (3.15) satisfy: |f(u)| ≤ α1|u|, u ∈H1 0(Ω), ρ∈C1(R;R+), ρ0(t)≤ρ∗<1. Then the trivial solution to (3.15) is stable (resp. asymptotically stable) provided that m≥α1 λ1 +LH p2(1 −ρ∗)λ1resp. m > α1 λ1 +LH p2(1 −ρ∗)λ1.(3.18) Proof. We construct the Lyapunov functional V:R+×C∞(H1 0(Ω)) to (3.15) in the form V(t, φ) = |φ(0)|2+C 1−ρ∗Z0 −ρ(t)|φ(θ)|2dθ, where C > 0 is a constant to be determined later on. Denoting by V(t) = V(t, ut(·;τ, ϕ)), where ut(·;τ, ϕ) is the solution to problem (3.15) with initial value ϕ, we have V(t) = |u(t)|2+C 1−ρ∗Z0 −ρ(t)|u(t+θ)|2dθ =|u(t)|2+C 1−ρ∗Zt t−ρ(t)|u(s)|2ds. Consequently, d dtV(t) = 2 < a(l(u))∆u, u > +2(f(u) + H(u(t−ρ(t))), u(t)) +C 1−ρ∗|u(t)|2−C(1 −ρ0(t)) 1−ρ∗|u(t−ρ(t))|2 ≤ −2mku(t)k2+2α1 λ1ku(t)k2+2LH √λ1|u(t−ρ(t))|ku(t)k +C (1 −ρ∗)λ1ku(t)k2−C|u(t−ρ(t))|2 ≤2−m+α1 λ1 +C 2(1 −ρ∗)λ1ku(t)k2+L2 H λ1ε|u(t−ρ(t))|2 +εku(t)k2−C|u(t−ρ(t))|2, 24 where the Poincar´e and Young inequalities have been used. Pick up ε=L2 H/λ1Cso that the term containing |u(t−ρ(t))|2vanishes in the above expression. Then we conclude that d dtV(t)≤2−m+α1 λ1 +C 2(1 −ρ∗)λ1 +L2 H 2λ1Cku(t)k2, and minimizing the coefficient in brackets in the right-hand side, which is attained for C= LH√1−ρ∗, we deduce d dtV(t)≤2 −m+α1 λ1 +LH 2p(1 −ρ∗)λ1!ku(t)k2. Then, the coefficient in the brackets in the right-hand side above is less than or equal or strictly less than zero depending on the conditions in (3.18). This, jointly with the Poincar´e inequality, implies the good control of d dt V(t) in order to apply Proposition 3.8. Therefore, both stability and asymptotic stability statements follow respectively.  4 Existence of pullback attractors The purpose of this section is to prove some general results on the global asymptotic behavior of problem (2.1) completing those shown in the previous sections. In order to proceed, we start with some standard notations related to dynamical systems. Definition 4.1 Given a metric space (X, d), a process Uon Xis a biparametric family of mappings U(t, τ) : X→Xfor −∞ < τ ≤t < +∞, with the following properties: (i)U(t, τ)∈C(X;X)for all t≥τ; (ii)U(τ, τ) = Id (the identity map) for all τ∈R; (iii)U(t, τ) = U(t, r)U(r, τ)for all −∞ < τ ≤r≤t < +∞. Next, we recall some useful concepts in order to study the asymptotic behavior of a process. Definition 4.2 For a process Udefined on a metric space (X, d), a family ˆ B0={B0(t) : t∈R} of subsets of Xis said to be pullback absorbing for bounded sets, if, for any bounded set Bof X, and any t, there exists a time T(B, t)such that, U(t, τ)B⊂B0(t),∀τ≤T(B, t). The process Uis said to be ˆ B0-pullback asymptotically compact if, for any t, and any sequences {τn},{xn} ⊂ Xwith τn≤t,limn→+∞τn=−∞, and xn∈B0(τn), the sequence {U(t, τn)xn}is relatively compact in X. Definition 4.3 A family A={A(t) : t∈R}is said to be a pullback attractor for a process Uif: (i)Each A(t)is a compact subset of Xfor all t∈R; 25 Integrating (4.13) between τand t, together with the above expression and (4.11)-(4.12), we have eκt|u(t)−u∗|2≤ |ϕ(0) −u∗|2eκτ +K(ρ)eκτ kϕ−u∗k2 L2((−ρ,0];L2(Ω)) +κZt τ eκs|u(s)−u∗|2ds +Zt τ eκs −mλ1+1 mL2 a|l|2ku∗k2+ 2Lf+1+K(ρ)|u(s)−u∗|2ds, Notice, if κis small enough, combine with assumption (4.12), the sum of last two terms of the above inequality keeps being negative. Therefore, |u(t)−u∗|2≤e−κ(t−τ)|ϕ(0) −u∗|2+K(ρ)kϕ−u∗k2 L2((−ρ,0];L2(Ω)). This finishes the proof.  Theorem 4.12 Assume the same conditions of Lemma 4.11. Then the process Upossesses a pullback attractor A={u∗}, which is a singleton. Moreover, it is also a forward attractor. Proof. From Lemma 4.11, it follows there exists a unique stationary solution u∗, such that for all τ∈R,∀ϕ∈L2((−ρ, 0]; L2(Ω)), the corresponding solution, u(·) = u(·;τ, ϕ) of problem (2.1) satisfies |u(t)−u∗|2≤e−κ(t−τ)|ϕ(0) −u∗|2+K(ρ)kϕ−u∗k2 L2((−ρ,0];L2(Ω)),(4.14) which implies, for each fixed t, and any bounded set ϕ∈B, dist(U(t, τ))B, u∗)→0 as τ→ −∞. Therefore, {u∗}attracts every bounded set of C([−ρ, 0]; L2(Ω)), thanks to the property of minimal attractor (Remark 4.5), A(t)⊂ {u∗}, immediately, A(t) = {u∗}. Further, estimation (4.14) also guarantees that {u∗}is forward attracting for every bounded set in C([−ρ, 0]; L2(Ω)), and then it is a forward attractor.  4.2 Pullback attractors in L2-norm in the unbounded delay case There are several phase spaces which allow us to deal with unbounded delay. For example, in the previous 3 sections, we have proved all of results in the phase space C∞(L2(Ω)). However, using this phase space, we are not able to prove asymptotic compactness of process U. To overcome this difficulty, here, we will consider the space Eγ={ϕ∈C((−∞,0]; L2(Ω)) : ∃lim s→−∞ eγsϕ(s)∈L2(Ω)},∀γ > 0, which is a Banach space endowed with the norm kϕkγ:= sup s∈(−∞,0] eγs|ϕ(s)|. 32 Therefore, our process U:R2 d×Eγ→Eγis defined by U(t, τ)ϕ=ut(·;τ, ϕ)∀ϕ∈Eγ,∀τ≤t. (4.15) With the help of the exponential appearing in this phase space, asymptotic compactness of the process Ucan be presented. Also, the global existence of weak solution in time of problem (2.1) with unbounded delay in Eγcan be proved under appropriate conditions. Proposition 4.13 Assume that function ais locally Lipschitz and (2.2) holds, f∈C(R)satisfying (2.4) and (4.3), l∈L2(Ω). Also suppose h:R×Eγ→L2(Ω), fulfilling (H2)-(H3)with X=Y=L2(Ω) (but slightly modifying condition (H3), i.e., replace Cρ(X)with Eγ(X)). Let λ:= λ1m−4α > 0,2γ > λ > L2 h mλ1 .(4.16) Then, the following estimations hold for any solution u(·) = u(·;τ, ϕ)to problem (2.1) for all t≥τ, kutk2 γ≤2e−(λ−L2 h mλ1)(t−τ)kϕk2 γ+2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h),(4.17) and m 2Zt τku(s)k2ds ≤e−λ(t−τ)|u(τ)|2+ 2e−(λ−L2 h mλ1)(t−τ)kϕk2 γ +2β|Ω| λeλ(t−τ)+L2 h λλ1m2β|Ω| λ+2L2 hβ|Ω| λ(λ1mλ −L2 h)eλ(t−τ). (4.18) Proof. Multiply (2.1) by u(t), by (2.2), (4.3), (H2)-(H3), the Young and Poincar´e inequalities, we have d dt|u(t)|2+λ1m 2−2α|u(t)|2+m 2ku(t)k2≤2β|Ω|+L2 h mλ1kutk2 γ, integrating between τand t, by (4.16), it follows |u(t)|2+m 2Zt τ e−λ(t−s)ku(s)k2ds ≤ |u(τ)|2e−λ(t−τ)+2β|Ω| λe−λ(t−τ) +L2 h mλ1Zt τ e−λ(t−s)kusk2 γds. (4.19) Further, neglecting the second term on the left hand side of (4.19), kutk2 γ≤maxsup θ∈(−∞,τ−t] e2γθ|ϕ(θ+τ−t)|2,sup θ∈[τ−t,0] e2γθ−λ(t+θ−τ)|u(τ)|2 +2β|Ω| λe2γθ−λ(t+θ−τ)+L2 h mλ1Zt+θ τ e2γθ−λ(t+θ−s)kusk2 γds. By assumption (4.16), on the one hand, sup θ∈(−∞,τ−t] e2γθ|ϕ(t+θ−τ)|2= sup θ≤0 e2γ(θ−(t−τ))|ϕ(θ)|2=e−2γ(t−τ)kϕk2 γ≤e−λ(t−τ)kϕk2 γ. 33 On the other hand, by the same reasoning as above, we obtain sup θ∈[τ−t,0] e2γθ−λ(t+θ−τ)|u(τ)|2≤e−λ(t−τ)|u(τ)|2,sup θ∈[τ−t,0] e2γθ−λ(t+θ−τ)2β|Ω| λ≤2β|Ω| λ, and sup θ∈[τ−t,0] L2 h mλ1Zt+θ τ e2γθ−λ(t+θ−s)kusk2 γds ≤L2 h mλ1Zt τ e−λ(t−s)kusk2 γds. Collecting these estimations, we deduce that kutk2 γ≤e−λ(t−τ)kϕk2 γ+2β|Ω| λ+L2 h mλ1Zt τ e−λ(t−s)kusk2 γds. By the Gronwall lemma, we have kutk2 γ≤2e−(λ−L2 h mλ1)(t−τ)kϕk2 γ+2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h),(4.20) namely, (4.17) is true. Now we will prove (4.18), in fact, in the middle of above manipulations, we have omitted a positive term in the left-hand side of (4.19), namely m 2Rt τeλ(t−s)kusk2ds. Indeed, combining (4.19) and (4.20), we have m 2Zt τ e−λ(t−s)ku(s)k2ds ≤ |u(τ)|2e−λ(t−τ)+2β|Ω| λe−λ(t−τ) +L2 h mλ1Zt τ2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h)+ 2e−(λ−L2 h mλ1)(s−τ)e−λ(t−s)kϕk2 γds, thus, (4.18) is proved. The proof of this proposition is complete.  Corollary 4.14 Under the assumptions of Proposition 4.13, the family ˆ B0={B0(t), t ∈R}with B0(t) = BEγ(0, %), where %2= 1 + 2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h), is pullback absorbing bounded sets for the process U. Proposition 4.15 Under assumptions of Corollary 4.14, the process Uis ˆ B0-asymptotically compact. Proof. Our proof relies on the energy method. Let t0∈R,un(·) be a sequence of solutions in their respective intervals [τn, t0], with initial data ϕn∈B0=BEγ(0, %), where τn→ −∞ as n→+∞. Consider the sequence ξn=un t0. Then we will prove that this sequence is relatively compact in Eγ. 34 Step 1: Consider two arbitrary values 0 < T < ¯ T, we will prove that ξn|[−¯ T,0] is relatively compact in C([−¯ T, 0]; L2(Ω)). It follows from (4.17), that there exists n0∈(t0, T ] such that for all n≥n0(t0, T), τn≤t0−Tand kun tk2 γ≤R, ∀t∈[t0−T, t0],∀n≥n0(t0, T), where R:= 1 + 2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h), therefore |un(t)|2≤R, ∀t∈[t0−T, t0],∀n≥n0(t0, T), kun t0−Tk2 γ≤R, ∀n≥n0(t0, T).(4.21) Let yn(·) = un(·+t0−T). Then, for each n≥1 such that τn< t0−T, the function yn(·) is a solution on [0, T] of a similar problem to (2.1), namely, hreplaced by ˜ h(s, ·) = h(s+t0−T, ·) and with yn 0=un t0−T,yn T=un t0=ξn. Then kynkγsatisfies the estimations in (4.21), for all n≥n0(t0, T). From (4.18) we have kynk2 L2([0,T];H1 0(Ω)) ≤K(t0, T). Hence, {yn}is bounded in L∞((0, T]; L2(Ω)) and L2((0, T]; H1 0(Ω)), and {(yn)0}is bounded in L2((0, T]; H−1(Ω)). Thus, up to a subsequence (relabelled the same), there exists a function y(·) such that                    yn→yweakly star in L∞((0, T ]; L2(Ω)), yn→yweakly in L2((0, T ]; H1 0(Ω)), (yn)0→y0weakly in L2((0, T ]; H−1(Ω)), yn→ystrongly in L2((0, T]; L2(Ω)), yn(t)→y(t) strongly in L2(Ω),a.e. t∈[0, T]. Moreover, reasoning as in the proof of Lemma 4.8, we claim for any sequence {tn},t0∈[0, T] with tn→t0, yn(tn)→y(t0) weakly in L2(Ω). Besides, by (H3) and (4.21), we obtain Zt 0|˜ h(s, yn s)|2ds ≤L2 hZt 0kyn sk2 γds ≤Ct, where Cdoes not depend on nor t, therefore, ˜ h(t, yn t)→υh(t) weakly in L2((0, T ]; L2(Ω)), 35 also Zt s|˜ h(r, yn r)|2dr ≤C(t−s), Zt s|υh(r)|2dr ≤lim inf n→∞ Zt s|˜ h(r, yn r)|2dr ≤C(t−s),∀0≤s≤t≤T. Then, in a standard way, one can prove y(·) is the unique weak solution to the problem          ∂y ∂t −a(l(y))∆y=f(y) + υh(t), y(x, t) = 0, y(x, 0) = y0(x), in Ω ×[0, T], on ∂Ω×[0, T], in Ω. (4.22) Multiplying (2.1) by un, (4.22) by y, and integrating between sto t, we obtain the energy inequality, 1 2|z(t)|2+mZt skz(r)k2dr ≤1 2|z(s)|2+Zt s (f(z(r), z(r))dr +C(t−s),0≤s≤t≤T, where z=ynor z=y. Then, the mappings Jn,J: [0, T]→Rdefined as J(t) = 1 2|y(t)|2+Zt 0 (f(y(r), y(r))dr −Ct, Jn(t) = 1 2|yn(t)|2+Zt 0 (f(yn(r), yn(r))dr −Ct, are non-increasing and continuous. Analogously to the proof of Lemma 4.8, for a fixed t0>0, using a sequence {tn}with tn%t0, we are able to establish the convergence of the norms, and therefore, jointly with the weak convergence, deduce that yn→yin C([δ, T]; L2(Ω)), for any δ > 0. Now, since T > ¯ T, we obtain that ξn→ϕin C([−¯ T, 0]; L2(Ω)), where ϕ(s) = y(s+T), for s∈[−¯ T, 0]. Repeating the same procedure for 2 ¯ T, 3 ¯ T,···, and using a diagonal argument, we obtain a continuous function ϕ: (−∞,0] →L2(Ω) and a subsequence such that ξn→ϕin C([−¯ T, 0]; L2(Ω)) on every interval [−¯ T, 0]. Moreover, since, for a fixed T > 0, un(s+t0), with s∈[−T, 0], satisfies the estimation (4.21) for any n≥n0(t0, T ), it is clear that |ϕ(t)|2≤1 + 2β|Ω| λ+2L2 hβ|Ω| λ(λmλ1−L2 h).(4.23) Step 2:We claim that ξnconverges to ϕin Eγ. Indeed, we have to see that for every ε > 0, there exists nεsuch that sup s∈(−∞,0] |ξn(s)−ϕ(s)|2e2γs ≤ε, ∀n≥nε.(4.24) 36 Fix Tε>0 such that e−2γTε≤ε 4, and take nε≥n0(t0, Tε) such that |ξn(s)−ϕ(s)|2e2γs ≤ε 2,∀s∈[−Tε,0] and τn≤t0−Tε,∀n≥nε. (This is possible since the convergence of ξnto ϕholds in compact intervals of time.) Thus, to prove (4.24), we only need to check sup s∈(−∞,−Tε]|ξn(s)−ϕ(s)|2e2γs ≤ε 2,∀n≥nε. By (4.23) and the choice of Tε, it is not difficult to check that, for all k∈N∪ {0}, and for all s∈[−(Tε+k+ 1),−(Tε+k)], it holds that e2γs|ϕ(s)|2≤e−2γ(Tε+k)|ϕ(s)|2≤ε 4. Hence, it suffices to prove the following sup s∈(−∞,−Tε] e2γs|ξn(s)|2≤ε 4,∀n≥nε. Recall that ξnhas two parts ξn(s) =    ϕn(s+t0−τn), s ∈(−∞, τn−t0], un(s+t0), s ∈[τn−t0,0]. Thus, the proof is finished if we prove that max sup s∈(−∞,τn−t0] e2γs|ϕn(s+t0−τn)|2,sup s∈[τn−t0,−Tε] e2γs|un(s+t0)|2≤ε 4. The first term above can be estimated as follows, sup s∈(−∞,τn−t0] e2γs|ϕn(s+t0−τn)|2 = sup s≤τn−t0 e2γ(s+t0−τn)e−2γ(t0−τn)|ϕn(s+t0−τn)|2 =e−2γ(t0−τn)kϕk2 γ ≤ε 4, thanks to the chose of nε. Finally, for the second term, we have sup s∈[τn−t0,−Tε] e2γs|un(s+t0)|2 = sup θ∈[τn−t0+Tε,0] e2γ(θ−Tε)|un(t0+θ−Tε)|2 ≤e−2γTεkun t0−Tεk2 γ ≤ε 4, where we have used (4.21) with T=Tε. The proof is complete.  37 Theorem 4.16 Assume that function ais locally Lipschitz and (2.2) holds, f∈C(R)satisfying (2.4) and (4.3), l∈L2(Ω). Also suppose h:R×Eγ→L2(Ω), fulfilling (H2)-(H3)with X=Y= L2(Ω) (but slightly modifying condition (H3), i.e., replacing Cρ(X)by Eγ(X)). Let λ:= λ1m−4α > 0,2γ > λ > L2 h mλ1 . Then the process Udefined in Eγassociated to (2.1) has a pullback attractor ˆ A={A(t)}. Remark 4.17 We end up this section with a comment on possible extension of our results. Instead of imposing condition (H3)on the delay term, if h(·,·)satisfies: (H0 3)There exists a function Lh(·)∈L1 loc(R), such that for all t∈(τ, T],α,β∈Cρ(X), kh(t, α)−h(t, β)kY≤Lh(t)kα−βkCρ(X). Then, all results in Section 4 still hold true but replacing the universe of bounded sets (to be attracted) by another appropriate universe of time-dependent families (e.g., see [25] for more details concerning this possibility). 5 Acknowledgements The research of the first author has been partially supported by CSC (China Scholarship Council, No. 201906280198) for 1 year’s study abroad at the University of Sevilla. The research of the second author has been partially supported by the National Natural Science Foundation of China (Nos. 11371286, 11401458) and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2019JM-165). 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