Existence of D−Pullback Attractor for an Infinite Dimensional Dynamical System

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2021036 DYNAMICAL SYSTEMS SERIES B

EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN INFINITE DIMENSIONAL DYNAMICAL SYSTEM

Mustapha Yebdri Laboratory of dynamical systems Department of Mathematics,Faculty of sciences University of Tlemcen Tlemcen, BP.119, 13000 Algeria

(Communicated by Jos´eA. Langa)

Abstract. At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system. For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs. In this paper we give a positive aftermath by proving the existence of pullback D-attractor. The key point is to find a bounded family of pullback D- absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback w-D-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.

1. Introduction. Many phenomena in the applied sciences can be described by a partial diﬀerential equations (PDEs) of the form ∂ ∂ u(t, x)−∆ u(t, x)−∆u(t, x)+f(u(t, x)) = b(t, u )(x)+g(t, x) in (τ, ∞)×Ω, (1) ∂t ∂t t where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, compli- mented by initial and boundary conditions   u = 0 on (τ, ∞) × ∂Ω u(τ, x) = u0(x), τ ∈ R and x ∈ Ω (2)  u(τ + θ, x) = ϕ(θ, x), θ ∈ [−r, 0] and x ∈ Ω.

2020 Mathematics Subject Classification. Primary: 35B41, 37L30. Key words and phrases. Infinite dimenisional dynamical systems, partial differential equations with delay, asymptotic behavior,pullback attractor, ω−limit set.

1 2 MUSTAPHA YEBDRI

∂u The equation (1) without the term ∆ ∂t , is a classical equation with delay. For more details on diﬀerential equations with delay we refer the reader to J.K. Hale [10] and J. Wu [20]. The study of the asymptotic behavior of such equations is of hudge importance. It is essential for practical applications to understand and even predict the long time behavior of the solutions of such equations. Assuming that the above Cauchy problem is well-posed we can deﬁne a family of solution operators

U(t, τ): X → X , 0 0 (u , ϕ) 7→ U(t, τ)(u , ϕ) = (u(t), ut)

U(t, τ), t ≥ τ, τ ∈ R, acting on some convenient space X (called the phase space), where u is the weak solution of (1)-(2). It is easy to see that this family of operators satisfies U(τ, τ) = Id, U(t, s) ◦ U(s, τ) = U(t, τ), t ≥ s ≥ τ, τ ∈ R, where Id denotes the identity operator. The pair (U(t, τ),X) is called the dynamical system associated with the problem (1)-(2). The qualitative study of finite dimensional dynamical systems goes back to the pioneering works of Poincaréon the N-body problem in the beginning of the 20th century (see, e.g., [2] and the references therein ). In particular, it was discovered, at the very beginning of the theory, that even relatively simple systems even of ordinary differential equations can generate very complicated (chaotic) behaviors. Furthermore, these systems are extremely sensitive to perturbations, in the sense that trajectories with close, but different, initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space having a very complicated geometric structure, which thus accumulates the nontrivial dynamics of the system, the so-called strange attractor (see, e.g.,[1][3][4][9][12]). For a distributed system whose initial state is described by functions depending on the spatial variable, the time evolution is usually governed by a system of partial differential equations (PDEs). In this case, the phase space X is (a subset of) an infinite dimensional function space; typically, X = L2(Ω) or L∞(Ω), where Ω is some domain in Rn. We will speak of infinite dimensional dynamical systems. An important difference with ODEs, is that the analytical structure of a PDE is much more complicated. In particular, we do not have a unique solvability result in general. The global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics. It is therefore natural to investigate whether the features mentioned above for dynamical systems generated by systems of ODEs, strange attractors, generalize to systems of PDEs. It is worth emphasizing once more that the phase space is an infinite dimensional function space. However, experiments showed that, similar to the case of finite dimensional dynamical systems, the trajectories are localized, up to some transient process, in a “thin” invariant subset of the phase space having a very complicated geometric structure, which thus accumulates all the essential dynamics of the system. From a mathematical point of view, this led to the notions of attractors (see [1],[5],[8],[12], [16], [17] and the references therein ). There are two important approaches that were developed in order to study the asymptotic behavior of non-autonomous differential equations. First, the theory of uniform attractor, a minimal compact (not invariant) set that forward attracts bounded sets uniformly with respect to the initial time. Second, the theory of EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 3 pullback attractor, a family of compact sets which are invariant and pullback (but, in general, not forward) attract bounded sets. These two approaches were treated, at first, as unrelated notions. However, in [6] the authors explore both notions, and using the skew-product semiflow associated to the equation, important relationships between the two notions were proved. It is well known that the compact Sobolev embedding can be applied to obtain the existence of pullback D-attractor as the solution of the equation has higher regularity, e.g., although the initial conditions only belong to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. In [13] the authors proved that If there exists a family of pullback D-absorbing sets for a norm-to-weak continuous process {U(t, τ)}, which is pullback w-D-limit compact, then it has a pullback D-attractor {A(t): t ∈ R}. ∂u However, since the equation (1) contains the term ∆ ∂t , the solution has no higher regularity and this is similar to hyperbolic case. In this paper, we prove the existence of pullback D-attractor. It is well known that for the existence of pullback D-attractors, the key point is to find a bounded family of pullback D-absorbing sets then the pullback w-D-limit compactness for the process corresponding to the ∂u solution of our problem. As noticed before, because of the term ∆ ∂t , the pullback w-D-limit compactness for the process can not be proved by the compact Sobolev embedding. The nonlinearity with critical exponent makes also some barriers. The novelty in this paper is how to overcome these difficulties? To answer the question, we apply the decomposition techniques and a method used in [19] to verify the pullback w-D-limit compactness of the process with delay. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions. The organization of the paper is as follows, first we specify the notations and the assumptions used in the paper, next we construct the continuous process. Then we prove the main result, which is the existence of pullback D-attractor. The key point is to find a bounded family of pullback D-absorbing sets then we apply the decomposition techniques and a method used in [19] to verify the pullback w-D-limit compactness, which it is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.

2. Notation and assumptions. In this section we introduce notations and assumptions concerning problem (1)-(2)

2.1. Deﬁnition of a pullback D-attractor. Let (X, d) be a complete metric space. Let us denote P(Y ) the family of all nonempty subsets of X, and suppose D is a nonempty class of parametrized sets Db = {D(t): t ∈ R} ⊂ P(X) . A family Ab = {A(t): t ∈ R} ⊂ P(Y ) is said to be a pullback D-attractor for {U(t, τ)} if 1. A(t) is compact for all t ∈ R ; 2. Ab is invariant; i.e., U(t, τ)A(τ) = A(t) , for all t ≥ τ ; 3. Ab is pullback D-attracting ; i.e., lim dist(U(t, τ)D(τ),A(t)) = 0 , τ→−∞

for all D ∈ Db and all t ∈ R ; 4. If {C(t): t ∈ R} is another family of closed attracting sets then A(t) ⊂ C(t) , for all t ∈ R . 4 MUSTAPHA YEBDRI where dist denotes the Hausdorﬀ semi-distance between sets ( dist(A, B) := supa∈A infb∈B d(a, b), where A, B are two subsets of X). In [13] the authors proved that if there exists a family of pullback D-absorbing sets Bb = {B(t): t ∈ R} ⊂ Db for a norm-to-weak continuous process {U(t, τ)}, which is pullback w-D-limit compact, then it has a pullback D-attractor {A(t): t ∈ R} given by \ [ A(t) = w(B,b t) = U(t, τ)B(τ) . s≤t τ≤s

2.2. Notations and assumptions. Let Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary ∂Ω , k.k and < ., . > be the norm and the inner product in L2(Ω) =: H respectively, k∇.k and h∇., ∇.i be the norm and the inner product of 1 2 H0 (Ω) =: V in L (Ω) respectively. We will use as norm k.kY , in the Banach space Y. Let λ1 > 0 be the ﬁrst eigenvalue of −∆ in Ω with the homogeneous Dirichlet n 2N 4 o condition. Let α ∈ (0, min N−2 , 3 + N ) and let c1, c2, c3, c4, k be positive con- R u stants, we assume that λ1 > max{c1, c3} . Let F (u) = 0 f(s)ds. To study Problem (1)-(2), we shall assume that 2 f(u)u ≥ −c1u − c2, (f1) 0 f (u) ≥ −c3, f(0) = 0, (f2) |f(u)| ≤ k(1 + |u|α), (f3) uf(u) − c F (u) lim inf 4 ≥ 0, (f4) |u|→∞ u2 F (u) lim inf ≥ 0. (f5) |u|→∞ u2 0 We infer from (f4) and (f5) that for any δ > 0 there exist positive constants cδ , cδ such that 2 uf(u) − c4F (u) + δu + cδ ≥ 0 , ∀u ∈ R , (f6) 2 0 F (u) + δu + cδ ≥ 0 , ∀u ∈ R . (f7) The operator b : R × L2([−r, 0]; L2(Ω)) → L2(Ω) satisﬁes : (bI) For all φ ∈ L2([−r, 0]; L2(Ω)) , the function R 3 t 7→ b(t, φ) is measurable and is in L2(Ω). (bII) b(t, 0) = 0 for all t ∈ R . 2 2 (bIII) ∃Lb > 0 such that ∀t ∈ R and ∀φ1, φ2 ∈ L ([−r, 0]; L (Ω))

kb(t, φ1) − b(t, φ2)k ≤ Lbkφ1 − φ2kL2([−r,0];L2(Ω)) . 2 2 (bIV) ∃Cb > 0 such that ∀t ≥ τ and ∀u, v ∈ L ([τ − r, t]; L (Ω)) Z t Z t 2 2 kb(s, us) − b(s, vs)k ds ≤ Cb ku(s) − v(s)k ds . τ τ−r Remark 1. From (bI)-(bIII), for T > τ and u ∈ L2([τ − r, T ]; L2(Ω)) the function ∞ 2 R 3 t 7→ b(t, ut) is measurable and belongs to L ((τ, T ); L (Ω)) . 2 2 0 1 The function g ∈ Lloc(R; L (Ω)) , u ∈ H0 (Ω) is the initial condition in τ and ϕ ∈ L2([−r, 0]; L2(Ω)) is also the initial condition in [τ − r, τ] , r > 0 is the length of the delay eﬀect. EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 5

3. Weak continuous process. 3.1. Existence and uniqueness of weak solutions. First, we define the concept of weak solutions. Definition 3.1. A function u ∈ L2([τ − r, T ]; L2(Ω)) is called a weak solution of the problem (1)-(2) if for all T > τ we have 1 u ∈ C([τ, T ]; H0 (Ω)) and ∂u ∈ L2([τ, T ]; H1(Ω)) , ∂t 0 1 1 with u(t) = ϕ(t−τ) , for t ∈ [τ−r, τ] , and for all test functions v ∈ C ([τ, T ]; H0 (Ω)) such that v(T ) = 0 , it satisfies Z T Z T Z ∂u Z T Z Z T Z − hu, v0i + ∇ ∇v + ∇u∇v + f(u)v τ τ Ω ∂t τ Ω τ Ω Z T Z T Z 0 = hb(t, ut), vi + gv + u , v(τ) . τ τ Ω 2 Consider {ek}k≥1 , the complete orthonormal eigenfunctions of −∆ in L (Ω) . 1 2 This is a basis of H0 (Ω) ∩ H (Ω). Consider m m X u (t) = γk,m(t)ek , m = 1, 2,... k=1 which is the approximate solutions of Faedo-Galerkin of order m , that is m m  du , e + ∆ ∂u , e + h∆um, e i + hf(um), e i = hb(t, um), e i + hg, e i  dt k ∂t k k k t k k m 0 0 m 0 1 hu (τ), eki = Pmu , ek = u , ek i.e. Pmu (τ) → u in H0 (Ω) m  hu (τ + θ), eki = hPmϕ(θ), eki = hϕ(θ), eki ∀θ ∈ (−r, 0) (3) m for all k = 1 . . . m . Here γk,m(t) = hu (t), eki denote the Fourier coefficients, 1 2 0 such that γk,m ∈ C ((τ, T ); R) ∩ L ((τ − r, T ), R) , γk,m(t) is absolutely continu- Pm 2 1 ous and Pmu(t) = k=1 hu, eki ek is the orthogonal projection of L (Ω) and H0 (Ω) in Vm = span{e1, . . . , em}. It is well-known that the above finite-dimensional de- layed system is well-posed (e.g. cf. [10]), at least locally, i.e. (3) is a system of differential functional differential equations for the functions γ1,m . . . γm,m . Thus, 2 m m for the initial conditions (ϕ, a) in L ((−r, 0); R ) × R , there exist tm > 0 and a m T m 2 m unique solution of (3), v (t) = (γ1,m(t) . . . γm,m(t)) with v ∈ L ((τ − r, τ); R ) m m m 1 m such that v |[τ−r,τ] = ϕ and v (τ) = a , and v |[τ,tm] ∈ C ([τ, tm]; R ) . Hence, the solution of (3) is defined on the interval [τ, tm] with τ < tm < T . The a priori estimates for the Faedo-Galerkin approximate solutions that we obtain will show that tm = T. m ∞ 1 Lemma 3.2. {u } is bounded in L ((τ, T ); H0 (Ω)) . Proof. Multiplying (1) by um and integrating over Ω , then using the hypothesis (f1) and the Cauchy inequality, we get 1 d kum(t)k2 + k∇um(t)k2 + k∇um(t)k2 − c kum(t)k2 − c |Ω| 2 dt 1 2 1 η 1 η0 ≤ kb(t, um)k2 + kum(t)k2 + kg(t)k2 + kum(t)k2 . 2η t 2 2η0 2 6 MUSTAPHA YEBDRI

1 2 By the continuous injection of H0 (Ω) in L (Ω) , we have 2 2 λ1kuk ≤ k∇uk . (4) Then, one obtains d kum(t)k2 + k∇um(t)k2 + (2λ − 2c − η − η0)kum(t)k2 dt 1 1 1 1 ≤ kb(t, um)k2 + kg(t)k2 + c |Ω| . η t η0 2 Integrating this last estimate over [τ, t] , t ≤ T we ﬁnd Z t m 2 m 2 0 m 2 m 2 ku (t)k + k∇u (t)k + (2λ1 − 2c1 − η − η ) ku (s)k ds ≤ ku (τ)k τ Z t Z t m 2 1 m 2 1 2 + k∇u (τ)k + kb(s, us )k ds + 0 kg(s)k ds + c2|Ω|(t − τ) . η τ η τ Therefore, by using (bIV), we obtain Z t m 2 m 2 0 m 2 ku (t)k + k∇u (t)k + (2λ1 − 2c1 − η − η ) ku (s)k ds τ C Z τ ≤ kum(τ)k2 + k∇um(τ)k2 + b kum(s)k2ds η τ−r Z t Z t Cb m 2 1 2 + ku (s)k ds + 0 kg(s)k ds + c2|Ω|(t − τ) . η τ η τ Thus Z t m 2 m 2 0 Cb m 2 ku (t)k + k∇u (t)k + 2λ1 − 2c1 − η − η − ku (s)k ds η τ C Z τ ≤ kum(τ)k2 + k∇um(τ)k2 + b kum(s)k2ds η τ−r Z t 1 2 + 0 kg(s)k ds + c2|Ω|(t − τ) . (5) η τ

0 Cb 2 2 Hence, when 2λ1 − 2c1 − η − η − η > 0 and g ∈ Lloc(R; L (Ω)) , one gets

m 2 m 2 m 2 Cb 2 k∇u (t)k ≤ ku (τ)k + k∇u (τ)k + kϕk 2 2 η L ([−r,0];L (Ω))

1 2 + kgk 2 2 + c |Ω|(t − τ) . (6) η0 L ([τ,t];L (Ω)) 2 It follows by this estimate that for all T > τ , we have

m ∞ 1 {u } is bounded in L ((τ, T ); H0 (Ω)) (7)

From (7), we deduce that the local solution um can be extended to the interval [τ, T ] .

∂ m 2 1 Lemma 3.3. ∂t u is bounded in L ((τ, T ); H0 (Ω)) . EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 7

dum Proof. Multiplying (1) by dt and integrating over Ω , we have 2 2 Z m d m d m 1 d m 2 m ∂u u (t) + ∇ u (t) + k∇u (t)k + f(u ) dt dt 2 dt Ω ∂t Z m Z m m ∂u ∂u = b(t, ut ) + g . (8) Ω ∂t Ω ∂t On the other hand, we have d dF ∂u F (u) = dt du ∂t ∂u = f(u) . ∂t So d Z Z ∂u F (u) = f(u) . dt Ω Ω ∂t Hence, (8) becomes 2 2 Z d m d m 1 d m 2 m u (t) + ∇ u (t) + k∇u (t)k + 2 F (u ) dt dt 2 dt Ω Z m Z m m ∂u ∂u = b(t, ut ) + g . Ω ∂t Ω ∂t Using the Young inequality, we ﬁnd 2 Z d m 1 d m 2 m 1 m 2 1 2 ∇ u (t) + k∇u (t)k + 2 F (u ) ≤ kb(t, ut )k + kg(t)k . dt 2 dt Ω 2 2 Integrating over [τ, t] , t ≤ T we get Z t 2 Z d m 1 m 2 m ∇ u (s) ds + k∇u (t)k + 2 F (u (t, x)) τ ds 2 Ω Z Z t 1 m 2 m 1 m 2 ≤ k∇u (τ)k + 2 F (u (τ, x)) + kb(s, us )k ds 2 Ω 2 τ 1 Z t + kg(s)k2ds . 2 τ By the hypothesis (bIV), we have Z t 2 Z d m 1 m 2 m ∇ u (s) ds + k∇u (t)k + 2 F (u (t, x)) τ ds 2 Ω 1 Z C Z t ≤ k∇um(τ)k2 + 2 F (um(τ, x)) + b kum(s)k2ds 2 Ω 2 τ−r 1 Z t + kg(s)k2ds 2 τ 1 Z C Z t ≤ k∇um(τ)k2 + 2 F (um(τ, x)) + b kum(s)k2ds 2 Ω 2 τ Z τ Cb m 2 1 2 + ku (s)k ds + kgkL2([τ,t];L2(Ω)) . 2 τ−r 2 8 MUSTAPHA YEBDRI

From this estimate and (5), we deduce that, for all T > τ ∂ um is bounded in L2((τ, T ); H1(Ω)) . (9) ∂t 0

m ∞ 1 Lemma 3.4 (Lemma 3.1 in [18]). If {u } is bounded in L ((τ, T ),H0 (Ω)) , then {f(um)} is bounded in Lq((τ, T ); Lq(Ω)) , (10) 2N+4 where q = αN . Now we can state the following general result of existence and uniqueness of solutions. 0 1 2 2 Theorem 3.5. For any τ ∈ R , T > τ , u ∈ H0 (Ω) , ϕ ∈ L ([−r, 0]; L (Ω)) and if 0 0 1 η+η Cb there exist small enough η , η < 2 such that λ1 > c1 + 2 + 2η , then the problem (1)(2) has a unique weak solution u on (τ, T ) . Proof. By (7), (9), (10), the hypothesis (bIV) and the remark1, we can extract a subsequence (relabelled the same) such that m ∞ 1 u * u weakly* in L ((τ, T ); H0 (Ω)), ∆um * ∆u weakly in L2((τ, T ); H−1(Ω)), ∂um ∂u * weakly in L2((τ, T ); H1(Ω)), ∂t ∂t 0 (11) ∂um ∂u ∆ * ∆ weakly in L2((τ, T ); H−1(Ω)), ∂t ∂t f(um) * σ0 weakly in Lq((τ, T ); Lq(Ω)), m 2 2 b(., u. ) → b(., u.) strongly in L ((τ, T ); L (Ω)) . Since f is continuous, we deduce that f(um) → f(u) a.e [τ, T ] × Ω . So from (10) and Lemma 1.3 page 12 in [14], we can identify σ0 with f(u) . Using the Aubin-Lions lemma of compactness, we deduce that um → u strongly in L2((τ, T ); L2(Ω)) . Thus um → u a.e [τ, T ] × Ω . Now, we have to prove that u(τ) = u0 . From (1), we have Z T Z T Z ∂u Z T Z Z T Z − hu, v0i + ∇ ∇v + ∇u∇v + f(u)v τ τ Ω ∂t τ Ω τ Ω Z T Z T Z = hb(t, ut), vi + gv + hu(τ), v(τ)i . (12) τ τ Ω In a similar way, from the Faedo-Galerkin approximations, we have Z T Z T Z ∂um Z T Z Z T Z − hum, v0i + ∇ ∇v + ∇um∇v + f(um)v τ τ Ω ∂t τ Ω τ Ω Z T Z T Z m m = hb(t, ut ), vi + gv+ < u (τ), v(τ) > . (13) τ τ Ω m 0 1 Using the fact that u (τ) → u in H0 (Ω) and (11) to ﬁnd Z T Z T Z ∂u Z T Z Z T Z − hu, v0i + ∇ ∇v + ∇u∇v + f(u)v τ τ Ω ∂t τ Ω τ Ω Z T Z T Z 0 = hb(t, ut), vi + gv + u , v(τ) . (14) τ τ Ω EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 9

Since v(τ) is given arbitrarily, comparing (12) and (14) we deduce that u(τ) = u0 . 1 m m To prove that u ∈ C([τ, T ]; H0 (Ω)) , we put w = u − u then we have ∂ ∂ wm − ∆ wm − ∆wm + f(um) − f(u) = b(t, um) − b(t, u ) . ∂t ∂t t t Multiplying this equation by wm and integrating over Ω , we obtain d Z kwm(t)k2 + k∇wm(t)k2 + 2k∇wm(t)k2 + 2 (f(um) − f(u)) wm dt Ω Z m m = 2 (b(t, ut ) − b(t, ut))(u − u) . Ω By the assumptions (f2) (bI) and (bIII), we get d kwm(t)k2 + k∇wm(t)k2 + 2k∇wm(t)k2 dt m 2 m 2 ≤ 2c3kw (t)k + 2Lbkwt kL2([−r,0];L2(Ω)) . Hence, by (4), one gets Z 0 d m 2 m 2 m 2 m 2 kw (t)k + k∇w (t)k ≤ 2c3kw (t)k + 2Lb kw (t + θ)k dθ dt −r Z 0 2c3 m 2 m 2 ≤ k∇w (t)k + 2Lb kw (t + θ)k dθ . λ1 −r Integrating over [τ, t] , we get kwm(t)k2 + k∇wm(t)k2 − kwm(τ)k2 + k∇wm(τ)k2 Z t Z t Z 0 2c3 m 2 m 2 ≤ k∇w (s)k ds + 2Lb kw (s + θ)k dθds λ1 τ τ −r Z t Z 0 Z t 2c3 m 2 m 2 ≤ k∇w (s)k ds + 2Lb kw (s)k dsdθ λ1 τ −r τ−r Z t Z τ Z t 2c3 m 2 m 2 m 2 ≤ k∇w (s)k ds + 2Lbr kw (s)k ds + 2Lbr kw (s)k ds . λ1 τ τ−r τ n o 2c3 Therefore we can ﬁnd β > max , 2Lbr such that λ1 Z t Z t 2c3 m 2 m 2 k∇w (s)k ds + 2Lbr kw (s)k ds λ1 τ τ Z t ≤ β k∇wm(s)k2 + kwm(s)k2 ds . τ So, we get kwm(t)k2 + k∇wm(t)k2 ≤ kwm(τ)k2 + k∇wm(τ)k2 Z τ Z t m 2 m 2 m 2 + 2Lbr kw (s)k ds + β k∇w (s)k + kw (s)k ds . τ−r τ Applying the Gronwall lemma to this estimate, we obtain kwm(t)k2 + k∇wm(t)k2 Z 0 m 2 m 2 m 2 β(t−τ) ≤ kw (τ)k + k∇w (τ)k + 2Lbr kw (τ + θ)k dθ e . (15) −r 10 MUSTAPHA YEBDRI

Since um(τ) → u0 and um(τ + θ) → ϕ(θ) , the estimate (15) shows that um → u 1 uniformly in C([τ, T ]; H0 (Ω)) . Finally, we prove the uniqueness and continuous dependence of the solution. To do this, we consider u1; u2 two solutions of problem (1) with the initial conditions u0,1, u0,2 and ϕ1, ϕ2 . Let w = u1 − u2. Similarly as in the proof of (15), we find kw(t)k2 + k∇w(t)k2 Z 0 (16) 2 2 2 β(t−τ) ≤ kw(τ)k + k∇w(τ)k + 2Lbr kw(τ + θ)k dθ e , −r and this completes the proof of this theorem. 3.2. Norm-to-weak Process. Definition 3.6. The two parameters family of operators U(τ, τ): X → X forms a norm-to-weak continuous process if 1. U(τ, τ)x = x , ∀τ ∈ R , x ∈ X ; 2. U(t, s)U(s, τ)x = U(t, τ)x , ∀t ≥ s ≥ τ , τ ∈ R , x ∈ X ; 3. U(t, τ)xn *U(t, τ)x if xn → x in X. The following result is useful for verifying that a process is norm-to-weak continuous. Proposition 1. [15] Let Y,Z be two Banach spaces, Y ∗,Z∗ be respectively their dual spaces. Assume that Y is dense in Z, the injection i : Y → Z is continuous and its adjoint i∗ : Z∗ → Y ∗ is dense, and {U(t, τ)} is a continuous or weak continuous process on Z. Then {U(t, τ)} is norm-to-weak continuous on Y iff for t ≥ τ , τ ∈ R ,U(t, τ) maps compact sets of Y into bounded sets of Y. 1 2 2 In what follows we consider the space X := H0 (Ω) × L ([−r, 0]; L (Ω)) , which is a Hilbert space with the norm Z 0 0 2 0 2 2 k(u , ϕ)kX = k∇u k + kϕ(θ)k dθ , −r as a phase space to the problem (1)-(2) As a consequence of the Theorem 3.5 we state 2 2 2 2 2 Corollary 1. We consider g ∈ Lloc(R; L (Ω)) , b : R × L ([−r, 0]; L (Ω)) → L (Ω) with the hypotheses (bI) − (bIV ) and f ∈ C1(R; R) verifying (f1)-(f5). Then the family of mappings U(t, τ): X → X 0 0 (u , ϕ) 7−→ U(t, τ)(u , ϕ) = (u(t), ut) , (17) with (t, τ) ∈ R2 and u is the weak solution to (1), defines a continuous process. In the following, we need to consider the Hilbert space 1 2 1 X1 = H0 (Ω) × L ([−r, 0]; H0 (Ω)) , with the norm Z 0 k(u0, ϕ)k2 = k∇u0k2 + k∇ϕ(θ)k2dθ . X1 −r

Remark 2. We remark that since t − τ ≥ r, U(t, τ) maps X to X1 . EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 11

4. Pullback D-attractors.

4.1. Pullback absorbing set.

Deﬁnition 4.1. [13] A family of bounded sets Bb = {B(t): t ∈ R} ⊂ Db is called pullback D-absorbing for the process {U(t, τ)} if for any t ∈ R and for any D ∈ Db , there exists τ0(t, D) ≤ t such that

U(t, τ)D(τ) ⊂ B(t) for all τ ≤ τ0(t, D) . Lemma 4.2. (Lemma 3.[11]) Assume that the function f satisﬁes (f3), (f6), (f7) and for all u, v ∈ L2([τ − r, t]; L2(Ω)) , b and g verify Z t Z t σs 2 σs 2 e kb(s, us) − b(s, vs)k ds ≤ Cb e ku(s) − v(s)k ds , (18) τ τ−r and Z t σs 2 e kg(s)k ds < ∞ , ∀t ∈ R , (19) −∞ n o 0 2λ1 where 0 < σ < δ < min 2c4, . Then for all t such that t ≥ τ + r and all 2λ1+1 (u0, ϕ) ∈ X, we have the following estimates

2N Z t 2 −σ(t−τ) 0 N−2 −σt σs 2 k∇u(t)k ≤ c e k(u , ϕ)kX + 1 + e e kg(s)k ds , (20) −∞ and

Z t 2N 2 −1 −σ(t−τ−r) 0 N−2 σr k∇u(s)k ds ≤ cµ e k(u , ϕ)kX + e t−r Z t + e−σ(t−r) eσskg(s)k2ds , (21) −∞ where µ := 4 δ0 − σ − Cb > 0 . λ1

Let R be the set of all functions ρ : R −→ (0, +∞) such that σt 2N lim e ρ N−2 (t) = 0 . t→−∞

By Db we denote the class of all families Db = {D(t): t ∈ R} ⊂ P(X) such that D(t) ⊂ BX (0, ρ(t)) , for some ρ ∈ R. Here BX (0, ρ(t)) denotes the closed ball in X centered at 0 with radius ρ(t) . Lemma 4.3 (Pullback D-absorbing set). Under the assumptions of lemma 4.2 and Z t −σt σs 2 ρ1(t) = c 1 + e e kg(s)k ds . −∞ Then, the family Bb given by 0 0 dφ 0 B(t) = (v , φ) ∈ X1 : k(v , φ)kX1 ≤ R(t), ≤ R (t) ; t ∈ R , ds L2([−r,0];H−1(Ω)) (22) with R(t) > 0 and R0(t) > 0 deﬁned respectively by 2 −1 σr R (t) = (1 + µ e )ρ1(t) , 12 MUSTAPHA YEBDRI

˜ 2 N N 02 λ(c0 + 2k) σr N−2 R (t) = 2 N−2 e N−2 ρ (t) 2 1 Cb −1 −1 2σr λ 2 + λλ µ e ρ (t) + kgk 2 2 , 2 1 1 2 L ([t−r,t];L (Ω)) is pullback D-absorbing for the process U(., .) deﬁned by (17) .

Proof. First, we observe that Bb ∈ D since for all t ∈ R , 0 0 B(t) ⊂ {(v , φ) ∈ X : k(v , φ)kX ≤ R(t)} , (23) with σt 2N lim e R N−2 (t) = 0 . t→−∞

Now, we will prove that U(t, τ)D(τ) ⊂ B(t) , for all τ ≤ τ0 . To do this, we proceed in two steps. Step 1. We will prove that: kU(t, τ)(u0, ϕ)k2 ≤ R2(t) . (24) X1 Fixed t ∈ R . By deﬁnition, we have Z t kU(t, τ)(u0, ϕ)k2 = k∇u(t)k2 + k∇u(s)k2ds . (25) X1 t−r From (21), for any t − r ≥ τ , we have

Z t 2N 2 −1 −σ(t−τ−r) 0 N−2 k∇u(s)k ds ≤ cµ e k(u , ϕ)kX t−r Z t + µ−1eσrc 1 + e−σt eσskg(s)k2ds , (26) −∞ for any (u0, ϕ) ∈ X. We substitute this inequality and (20) in (25) and by the deﬁntion of ρ1(t) , we get

2N kU(t, τ)(u0, ϕ)k2 ≤ ce−σ(t−τ)k(u0, ϕ)k N−2 1 + µ−1eσr + 1 + µ−1eσr ρ (t) X1 X 1 2N −σ(t−τ) 0 N−2 −1 σr 2 ≤ ce k(u , ϕ)kX 1 + µ e + R (t) , for all t − r ≥ τ and all (u0, ϕ) ∈ X. Since eστ → 0 when τ → −∞ , we obtain (24). Step 2. This step concerns the asymptotic estimate using R0(t) , we assume that ∂u t − 2r ≥ τ . Multiplying (1) by ∂t and integrating over Ω , after calculation we get 2 Z d 1 d 2 1 2 1 2 ∇ u(t) + k∇u(t)k + 2 F (u) ≤ kb(t, ut)k + kg(t)k . dt 2 dt Ω 2 2 1 −1 Since H0 (Ω) ⊂ H (Ω) with continuous embedding there exists λ > 0 such that 2 2 kukH−1(Ω) ≤ λk∇uk , one has 2 Z −1 d 1 d 2 1 2 1 2 λ u(t) + k∇u(t)k + 2 F (u) ≤ kb(t, ut)k + kg(t)k . dt H−1(Ω) 2 dt Ω 2 2 EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 13

Integrating over [t − r, t] , we ﬁnd

−1 R t d 2 1 2 R λ t−r ds u(s) H−1(Ω) ds + 2 k∇u(t)k + 2 Ω F (u(t)) 1 2 R ≤ 2 k∇u(t − r)k + 2 Ω F (u(t − r)) 1 R t 2 1 R t 2 + 2 t−r kg(s)k ds + 2 t−r kb(s, us)k ds . 2 2 From (bII) and (bIV) and using λ1kuk ≤ k∇uk , one has Z t Z t 2 2 kb(s, us)k ds ≤ Cb ku(s)k ds t−r t−2r Z t −1 2 ≤ Cbλ1 k∇u(s)k ds . t−2r By this estimate and (27), we get Z t 2 Z d λ 2 u(s) ds ≤ k∇u(t − r)k + 2 F (u(t − r)) t−r ds H−1(Ω) 2 Ω λ Z t C λλ−1 Z t + kg(s)k2ds + b 1 k∇u(s)k2ds . 2 t−r 2 t−2r

2N 1 N−2 2N By the embedding of H0 (Ω) in L (Ω) , (4) and the fact that 1 < N−2 , one has Z ˜0 ˜ kk k1 2N −1p 0 N−2 F (u) ≤ kλ1 |Ω|k∇uk + k∇uk Ω α + 1 ˜0 ˜ −1p 2N kk0k1 2N ≤ kk˜ λ |Ω|k∇uk N−2 + k∇uk N−2 2 1 α + 1 ˜0 ˜ ! −1p kk0k1 2N ≤ kk˜ λ |Ω| + k∇uk N−2 2 1 α + 1

2N ≤ k˜k∇uk N−2 , (27)

˜0 ˜ ˜ ˜ −1p kk0k1 where k := kk2λ1 |Ω| + α+1 .

2N By (27) and the fact that 2 < N−2 , we have Z t 2 d λ 2 2N u(s) ds ≤ k∇u(t − r)k + 2k˜k∇u(t − r)k N−2 t−r ds H−1(Ω) 2 λ Z t C λλ−1 Z t + kg(s)k2ds + b 1 k∇u(s)k2ds 2 t−r 2 t−2r ˜ λ(c0 + 2k) 2N λ N−2 2 ≤ k∇u(t − r)k + kgk 2 2 2 2 L ([t−r,t];L (Ω)) C λλ−1 Z t + b 1 k∇u(s)k2ds . (28) 2 t−2r We replace t by t − r in (20), we get

2N 2 −σ(t−r−τ) 0 N−2 k∇u(t − r)k ≤ ce k(u , ϕ)kX Z t−r + c 1 + e−σ(t−r) eσ(s−r)kg(s − r)k2ds . −∞ 14 MUSTAPHA YEBDRI

Because t − r ≤ t and eσr > 1 , we ﬁnd

2N 2 σr −σ(t−τ) 0 N−2 k∇u(t − r)k ≤ ce e k(u , ϕ)kX Z t + eσrc 1 + e−σt eσskg(s)k2ds −∞ 2N σr −σ(t−τ) 0 N−2 σr ≤ ce e k(u , ϕ)kX + e ρ1(t) . Hence

N N−2 2N 2N N−2 σr −σ(t−τ) 0 N−2 σr k∇u(t − r)k ≤ ce e k(u , ϕ)kX + e ρ1(t) .

N Since N−2 > 1 , we use the convexity to obtain

2N2 2N N N N N (N−2)2 N−2 N−2 −1 N−2 N−2 σr − N−2 σ(t−τ) 0 k∇u(t − r)k ≤ 2 c e e k(u , ϕ)kX N N N N−2 −1 σr N−2 N−2 +2 e ρ1 (t) . (29) Using (21), we replace r by 2r , so for any t − 2r ≥ τ , we ﬁnd

Z t 2N 2 −1 2σr −σ(t−τ) 0 N−2 −1 2σr k∇u(s)k ds ≤ cµ e e k(u , ϕ)kX + µ e ρ1(t) . (30) t−2r

We conclude from (28)-(30) that for all t − 2r ≥ τ and for any (u0, ϕ) ∈ X, we have

Z t 2 d u(s) ds t−r ds H−1(Ω) ˜ 2N2 λ(c0 + 2k) 2 N N σr − N σ(t−τ) 0 (N−2)2 ≤ 2 N−2 c N−2 e N−2 e N−2 k(u , ϕ)k 2 X

2N ˜ 2 N N Cb −1 −1 2σr −σ(t−τ) 0 N−2 λ(c0 + 2k) σr N−2 + λλ cµ e e k(u , ϕ)k + 2 N−2 e N−2 ρ (t) 2 1 X 2 1 Cb −1 −1 2σr λ 2 + λλ µ e ρ (t) + kgk 2 2 . 2 1 1 2 L ([t−r,t];L (Ω)) Hence, for all t − 2r ≥ τ and for any (u0, ϕ) ∈ X, we have

2 Z t 2N d Cb −1 −1 2σr −σ(t−τ) 0 N−2 u(s) ds ≤ λλ1 cµ e e k(u , ϕ)kX t−r ds H−1(Ω) 2 ˜ 2N2 λ(c0 + 2k) 2 N N σr − N σ(t−τ) 0 (N−2)2 02 + 2 N−2 c N−2 e N−2 e N−2 k(u , ϕ)k + R (t) . 2 X Since τ → −∞ , we have eστ → 0 and so we get

Z t 2 d 02 u(s) ds ≤ R (t) . (31) t−r ds

Therefore, it is clear to see from (24), (31) and the deﬁnition of Db , that the family Bb given by (22) is a pullback D-absorbing for the process U(., .) . EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 15

4.2. Pullback w-limit set. Deﬁnition 4.4. [13] A process {U(t, τ)} is called pullback w-D-limit compact if for all ε > 0 and D ∈ Db , there exists τ0(t, D) ≤ t such that   [ K  U(t, τ)D(τ) ≤ ε ,

τ≤τ0 where K(B) is the Kuratowski measure of noncompactness of B ∈ P(Y ) , i.e. K(B) = inf{δ > 0/B has a ﬁnite open cover of sets of diameter less or equals to δ} .

Now, we prove the following result

Proposition 2. Let {U(t, τ)} be a process on X, and let Bb = {B(t): t ∈ R} be a pullback D-absorbing set of {U(t, τ)} . Suppose that for each t ∈ R , any Bb ∈ D and any ε > 0 , there exist τ0 = τ0(t, B,b ε) ≤ t , a ﬁnite dimensional subspace Vn of 1 2 H0 (Ω) and L (Ω) , and a δ > 0 such that 1. for all τ ≤ τ0 , (u(t), ut) ∈ U(t, τ)B(τ) ,

kPn(u(t), ut)kX1 is bounded ;

2. for all τ ≤ τ0 , (u(t), ut) ∈ U(t, τ)B(τ) ,

k(I − Pn)(u(t), ut)kX1 < ε ;

3. for all τ ≤ τ0 , ut ∈ U(t, τ)B(τ) and for all l ∈ R with |l| < δ , we have

kPn(Tlut − ut)kL2([−r,0];H−1(Ω)) < ε .

Where Tlut is the translation (Tlut)(θ) = u(t + θ + l) with θ ∈ [−r, 0] and Pn is the canonical projection on Vn and I is the identity. Then {U(t, τ)} is pullback ω-D-limit compact in X with respect to each t ∈ R . Proof. (i) First, we will prove that {U(t, τ)} is pullback ω-D-limit compact in X1 . Note that by the properties of the measure K,((see the Lemma 1[13]) one has      [ [ K  U(t, τ)B(τ) ≤ K Pn  U(t, τ)B(τ)

τ≤τ0 τ≤τ0    [ + K (I − Pn)  U(t, τ)B(τ) . (32)

τ≤τ0 n o The assumption (1) gives that P S U(t, τ)B(τ) is contained in a ball n τ≤τ0 of ﬁnite radius . So we get    [ K Pn  U(t, τ)B(τ) ≤ K(B(0, ε0)) . (33)

τ≤τ0 By the properties of the measure K, one has

K(B(0, ε0)) ≤ 2ε0 . (34) 16 MUSTAPHA YEBDRI

Thus, by (33) and (34) it follows that    [ K Pn  U(t, τ)B(τ) ≤ 2ε0 . (35)

τ≤τ0 On the other hand, the assumption (2) and the properties of the measure K, give    [ K (I − Pn)  U(t, τ)B(τ) ≤ 2ε . (36)

τ≤τ0 Therefore, by (32), (35) and (36) we deduce that   [ 0 K  U(t, τ)B(τ) ≤ 2ε ,

τ≤τ0 0 where ε := ε0+ε , and this shows that {U(t, τ)} is pullback D-w-limit compact n0 0,n0 n0 n0 in X1 , i.e.; for all τ ≤ τ0 , any sequences τ → −∞ and (u , ϕ ) ∈ B(τ ) , n0 n0 n0 0,n0 n0 the sequence {(u (t), ut )} = {U(t, τ )(u , ϕ )} is relatively compact in X1 . 2 −1 (ii) Next, we will check the equicontinuity property of ut in L ([−r, 0]; H (Ω)) . To this end, we use the Lp-versions of Arzel`a-AscoliTheorem (see Theo- rem IV.25 in [7], p.72). The assumption (2) gives

k(I − P )u k 2 1 < ε . (37) n t L ([−r,0];H0 (Ω)) 1 −1 Since H0 (Ω) ⊂ H (Ω) with continuous injection, we have

k(I − P )u k 2 −1 ≤ c k(I − P )u k 2 1 . n t L ([−r,0];H (Ω)) 5 n t L ([−r,0];H0 (Ω)) So by this estimate and (37), one has 0 k(I − Pn)utkL2([−r,0];H−1(Ω)) < ε , (38) 0 where ε := c5ε . Hence from this estimate and the assumption (3), we deduce + that for all τ ≤ τ0 , ut ∈ U(t, τ)B(τ) and all l ∈ R with l < δ , we have

kTlut − utkL2([−r,0];H−1(Ω)) ≤ kPn(Tlut − ut)kL2([−r,0];H−1(Ω))

+ k(I − Pn)(Tlut − ut)kL2([−r,0];H−1(Ω)) < ε00 , with ε00 := ε + ε0 and this is the needed equicontinuity. n0 2 1 From (i), we deduce that {ut } is relatively compact in L ([−r, 0]; H0 (Ω)) , n0 2 −1 and (ii) gives that {ut } is relatively compact in L ([−r, 0]; H (Ω)) . It follows n0 2 2 that {ut } is relatively compact in L ([−r, 0]; L (Ω)) . Therefore, we conclude that {U(t, τ)} is pullback ω-D-limit compact in X, which completes the proof of the Proposition2. Theorem 4.5. The process {U(t, τ)} corresponding to (1)(2)has a pullback D- attractor Ab = {A(t): t ∈ R} in X. Proof. From Lemma 4.3, {U(t, τ)} has a family of Pullback D-absorbing sets in X. By the result proved in [13], it remains to show that {U(t, τ)} is Pullback w-D-limit compact. To this end, we need to check the conditions (1)-(3) in Proposition2. EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 17

To this aim, we decompose the nonlinearity f in the following form.

f = f0 + f1 , 1 where f0 , f1 ∈ C (R, R) satisfy 2 f0(u)u ≥ −c1u − c2, (39) 0 f0(u) ≥ −c3, (40) α |f0(u)| ≤ k(1 + |u| ) , (41) α |f1(u)| ≤ k(1 + |u| ) . (42) 1 Let λ1, λ2,... be the eigenvalues of −∆ in H0 (Ω) and w1, w2,... the corresponding eigenfunctions. Without loss of generality, we can assume that 0 < λ1 ≤ λ2 ≤ 2 ... ≤ λj as j → +∞ . Then {w1, w2,...} form an orthogonal basis in L (Ω) and 1 H0 (Ω) . Let Vn = span{w1, w2, . . . , wn} ,Pn be the canonical projection on Vn and 0 I be the identity. Then we can decompose U(t, τ)(u , ϕ) = (u(t), ut) as follows

u(t) = v(t) + w(t), ut = vt + wt . Here v and w solve the following equations  ∂ v − ∆ ∂ v − ∆v + f (v) = P b(t, v ) + P g  ∂t ∂t 0 n t n  v = 0 on ∂Ω 0 (43) v(τ, x) = Pnu   v(τ + θ, x) = Pnϕ(θ), θ ∈ [−r, 0] and  ∂ w − ∆ ∂ w − ∆w + f(u) − f (v) = b(t, u ) − P b(t, v ) + (I − P )g  ∂t ∂t 0 t n t n  w = 0 on ∂Ω 0 (44) w(τ, x) = (I − Pn)u   w(τ + θ, x) = (I − Pn)ϕ(θ), θ ∈ [−r, 0]

1. First, we will establish that for all τ ≤ τ0 , (u(t), ut) ∈ U(t, τ)B(τ) , we have kPn(u(t), ut)kX1 is bounded; to do this, we multiply (43) by v and integrating over Ω , in fact Z d 2 2 2 (kv(t)k + k∇v(t)k ) + 2k∇v(t)k + 2 f0(v)v dt Ω Z Z = 2 Pnb(t, vt)v + 2 Pngv . Ω Ω

Since Png ≤ g , Pnb ≤ b and by (39) and the Cauchy inequality one obtains d (kv(t)k2 + k∇v(t)k2) + 2k∇v(t)k2 dt 2 1 2 2 1 2 ≤ 2c1kv(t)k + 2c2|Ω| + kb(t, vt)k + (ε4 + ε5)kv(t)k + kg(t)k ε4 ε5 2 1 2 1 2 ≤ (2c1 + ε4 + ε5)kv(t)k + 2c2|Ω| + kb(t, vt)k + kg(t)k . ε4 ε5 Integrating from τ to t , for τ ≤ t ≤ T, we ﬁnd Z t kv(t)k2 + k∇v(t)k2 + 2 k∇v(s)k2ds τ Z t 2 2 2 ≤ kv(τ)k + k∇v(τ)k + (2c1 + ε4 + ε5) kv(s)k ds τ 18 MUSTAPHA YEBDRI

Z t Z t 1 2 1 2 + kb(s, vs)k ds + kg(s)k ds + 2c2|Ω|(t − τ) . ε4 τ ε5 τ By using (bII) and (bIV) we have Z t kv(t)k2 + k∇v(t)k2 + 2 k∇v(s)k2ds τ Z t 2 2 2 ≤ kv(τ)k + k∇v(τ)k + (2c1 + ε4 + ε5) kv(s)k ds τ Z t Z t Cb 2 1 2 + kv(s)k ds + kg(s)k ds + 2c2|Ω|(t − τ) ε4 τ ε5 τ Z t 2 2 2 ≤ kv(τ)k + k∇v(τ)k + (2c1 + ε4 + ε5) kv(s)k ds τ Z τ Z t Z t Cb 2 Cb 2 1 2 + kv(s)k ds + kv(s)k ds + kg(s)k ds + 2c2|Ω|(t − τ) . ε4 τ−r ε4 τ ε5 τ A direct computation gives Z t kv(t)k2 + k∇v(t)k2 + 2 k∇v(s)k2ds τ C Z τ ≤ kv(τ)k2 + k∇v(τ)k2 + b kv(s)k2ds ε4 τ−r Z t Z t Cb 2 1 2 + 2c1 + ε4 + ε5 + kv(s)k ds + kg(s)k ds + 2c2|Ω|(t − τ) . ε4 τ ε5 τ By (4), one obtains Z t kv(t)k2 + k∇v(t)k2 + 2 k∇v(s)k2ds τ C Z τ ≤ kv(τ)k2 + k∇v(τ)k2 + b kv(s)k2ds ε4 τ−r Cb t t 2c1 + ε4 + ε5 + Z Z ε4 2 1 2 + k∇v(s)k ds + kg(s)k ds + 2c2|Ω|(t − τ) . λ1 τ ε5 τ Thus, one ﬁnds

Cb ! t 2c1 + ε4 + ε5 + Z kv(t)k2 + k∇v(t)k2 + 2 − ε4 k∇v(s)k2ds λ1 τ C Z τ ≤ kv(τ)k2 + k∇v(τ)k2 + b kv(s)k2ds ε4 τ−r Z t 1 2 + kg(s)k ds + 2c2|Ω|(t − τ) . ε5 τ

Cb 2c1+ε4+ε5+ ε4 By the Theorem 3.5, we have η1 := 2 − > 0 . This last estimate gives λ1 C Z τ k∇v(t)k2 ≤ kv(τ)k2 + k∇v(τ)k2 + b kv(s)k2ds ε4 τ−r Z T 1 2 + kg(t)k dt + 2c2|Ω|(T − τ) , (45) ε5 τ EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 19 and Z t Z τ 2 −1 2 2 Cb 2 k∇v(s)k ds ≤ η1 kv(τ)k + k∇v(τ)k + kv(s)k ds τ ε4 τ−r Z T 1 2 + kg(t)k dt + 2c2|Ω|(T − τ) . (46) ε5 τ Therefore, for t − r ≥ τ we have [t − r, t] ⊂ [τ, t] and Z t Z t k∇v(s)k2ds ≤ k∇v(s)k2ds t−r τ Z τ −1 2 2 Cb 2 ≤ η1 kv(τ)k + k∇v(τ)k + kv(s)k ds ε4 τ−r −1 Z T η1 2 −1 + kg(t)k dt + 2c2η1 |Ω|(T − τ) . (47) ε5 τ We add (45) and (47) to get Z t k∇v(t)k2 + k∇v(s)k2ds t−r Z τ −1 2 2 Cb 2 ≤ (1 + η1 ) kv(τ)k + k∇v(τ)k + kv(s)k ds ε4 τ−r 1 2 2 + kgkL2([τ,T ];L2(Ω)) + 2c2|Ω|(T − τ) := M . (48) ε5 Hence, one obtains

kPn(u(t), ut)kX1 ≤ M, and this shows that the condition (1) in Proposition2 holds true. 2. Now, taking the inner product in L2(Ω) of (44) with w , we get Z d 2 2 2 kw(t)k + k∇w(t)k + 2k∇w(t)k + 2 (f(u) − f0(v)) w dt Ω Z Z = 2 (b(t, ut) − Pnb(t, vt)) w + 2 (I − Pn)g w . (49) Ω Ω

Since f0(v) = f(v) − f1(v) and Pnb ≤ b , (I − Pn)g ≤ g we obtain d Z kw(t)k2 + k∇w(t)k2 + 2k∇w(t)k2 + 2 |f(u) − f(v)||w| dt Ω Z Z Z + 2 |f1(v)||w| ≤ 2 |b(t, ut) − Pnb(t, vt)| |w| + 2 |(I − Pn)g| |w| Ω Ω Ω Z Z ≤ 2 |b(t, ut) − b(t, vt)| |w| + 2 |g| |w| . Ω Ω By (3), we have Z 2 (f(u) − f(v))(u − v) ≥ −c3ku − vk . (50) Ω Thus, by (50) and using the Cauchy inequality, (49) becomes Z d 2 2 2 2 kw(t)k + k∇w(t)k + 2k∇w(t)k ≤ 2c3kw(t)k + 2 |f1(v)| |w| dt Ω 2 1 2 2 1 2 + ε1kw(t)k + kb(t, ut) − b(t, vt)k + ε2kw(t)k + kg(t)k . (51) ε1 ε2 20 MUSTAPHA YEBDRI

By virtue of (42) and H¨olderinequality, we have Z Z α |f1(v)| |w| ≤ k (1 + |v| )|w| Ω Ω N+2 N−2 Z 2N Z 2N α 2N 2N ≤ k (1 + |v| ) N+2 |w| N−2 . Ω Ω By the convexity of power, one obtains

N+2 Z Z 2N N−2 α 2N |f1(v)| |w| ≤ 2 2N k 1 + |v| N+2 kwk 2N . N−2 Ω Ω L (Ω) n N+2 4 o 2N 2N Since α < min N−2 , 2 + N , one has α N+2 < N−2 for all N ≥ 3 . Then one ﬁnds

N+2 2N Z N−2 Z 2N |f1(v)| |w| ≤ 2 2N k 1 + k1|v| N−2 kwk 2N N−2 Ω Ω L (Ω) N+2 N−2 2N 2N N−2 2N ≤ 2 k |Ω| + k1kvk 2N kwk 2N . L N−2 (Ω) L N−2 (Ω) Similarly one gets

N+2 Z N−2 N+2 N+2 N+2 2N N−2 2N 2N 2N −1 2N 2N |f1(v)| |w| ≤ 2 2 k |Ω| + k kvk 2N kwk 2N 1 N−2 Ω L N−2 (Ω) L (Ω) N+2 N+2 N+2 N−2 2N 2N ≤ k |Ω| + k1 kvk 2N kwk 2N . L N−2 (Ω) L N−2 (Ω)

2N 1 N−2 0 By the embedding of H0 (Ω) in L (Ω), there exist positive constants k2 , c0 such that

Z N+2 N+2 N+2 2N 0 2N N−2 |f1(v)| |w| ≤ k2k |Ω| + c0k1 k∇vk k∇wk. Ω

n N+2 N+2 o 0 2N Hence, there exists k3 ≥ max k2k|Ω| 2N , k2kc0k1 such that

Z N+2 |f1(v)| |w| ≤ k3 1 + k∇vk N−2 k∇wk . Ω N+2 N+2 By (48), we have k∇vk N−2 ≤ M N−2 , then we get

Z N+2 |f1(v)| |w| ≤ k3 1 + M N−2 k∇wk . Ω Using the Cauchy inequality, one ﬁnds Z 2 2 k3 N+2 ε3 2 |f1(v)| |w| ≤ 1 + M N−2 + k∇w(t)k . Ω 2ε3 2 This last estimate and (51) lead to d kw(t)k2 + k∇w(t)k2 + (2 − ε )k∇w(t)k2 ≤ (2c + ε + ε )kw(t)k2 dt 3 3 1 2 2 2 1 2 1 2 k3 N+2 + kb(t, ut) − b(t, vt)k + kg(t)k + 1 + M N−2 . ε1 ε2 ε3 Using (4), one has d kw(t)k2 + k∇w(t)k2 + (2 − ε )k∇w(t)k2 dt 3 EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 21

2c3 + ε1 + ε2 2 1 2 ≤ k∇w(t)k + kb(t, ut) − b(t, vt)k λ1 ε1 2 2 1 2 k3 N+2 + kg(t)k + 1 + M N−2 . ε2 ε3

2c3+ε1+ε2 For λ1 > c3 , ε3 < 1 and ε1 , ε2 small enough, we have 2 − ε3 − > 0 . So λ1 one gets d 2 2 2c3 + ε1 + ε2 2 kw(t)k + k∇w(t)k + 2 − ε3 − k∇w(t)k dt λ1 2 2 1 2 1 2 k3 N+2 ≤ kb(t, ut) − b(t, vt)k + kg(t)k + 1 + M N−2 . ε1 ε2 ε3 Similarly as in the proof of Lemma 3.[11])(see Lemma 4.2) , we can choose a positive n o 0 2λ1 constant δ < min 2c4, such that 2λ1+1 0 2 2 2c3 + ε1 + ε2 2 δ (k∇w(t)k + kw(t)k ) ≤ 2 − ε3 − k∇w(t)k . λ1 In fact 2 2 d 0 1 2 1 2 k3 N+2 y(t) + δ y(t) ≤ kb(t, ut) − b(t, vt)k + kg(t)k + 1 + M N−2 , dt ε1 ε2 ε3 where y(t) = k∇w(t)k2 +kw(t)k2 . Multiplying this last inequality by eσt , such that σ < δ0 , to ﬁnd d eσt y(t) + δ0eσty(t) dt 2 2 σt 1 2 1 σt 2 k3 N+2 σt ≤ e kb(t, ut) − b(t, vt)k + e kg(t)k + 1 + M N−2 e . ε1 ε2 ε3 On the other hand, we have d d (eσty(t)) = σeσty(t) + eσt y(t) dt dt 0 σt σt 1 2 ≤ (σ − δ )e y(t) + e kb(t, ut) − b(t, vt)k ε1 2 2 1 σt 2 k3 N+2 σt + e kg(t)k + 1 + M N−2 e . ε2 ε3 Integrating it from τ to t , we obtain Z t y(t) ≤ e−σ(t−τ)y(τ) + (σ − δ0)e−σt eσsy(s)ds τ Z t 1 −σt σs 2 + e e kb(s, us) − b(s, vs)k ds ε1 τ Z t 2 2 Z t 1 −σt σs 2 k3 N+2 −σt σs + e e kg(s)k ds + 1 + M N−2 e e ds ε2 τ ε3 τ Z t ≤ e−σ(t−τ)y(τ) + (σ − δ0)e−σt eσsy(s)ds τ Z t 1 −σt σs 2 + e e kb(s, us) − b(s, vs)k ds ε1 τ Z t 2 2 1 −σt σs 2 k3 N+2 −σ(t−τ) + e e kg(s)k ds + 1 + M N−2 1 − e . ε2 τ ε3σ 22 MUSTAPHA YEBDRI

We use (18) and (bII) to get Z t y(t) ≤ e−σ(t−τ)y(τ) + (σ − δ0)e−σt eσsy(s)ds τ C Z t + b e−σt eσsku(s) − v(s)k2ds ε1 τ−r Z t 2 2 1 −σt σs 2 k3 N+2 −σ(t−τ) + e e kg(s)k ds + 1 + M N−2 1 − e ε2 τ ε3σ Z t ≤ e−σ(t−τ)y(τ) + (σ − δ0)e−σt eσsy(s)ds τ C Z τ C Z t + b e−σt eσskw(s)k2ds + b e−σt eσskw(s)k2ds ε1 τ−r ε1 τ Z t 2 2 1 −σt σs 2 k3 N+2 + e e kg(s)k ds + 1 + M N−2 . ε2 τ ε3σ Since Z t Z τ Z t eσsku(s)k2ds ≤ eστ ku(s)k2ds + eσsku(s)k2ds, τ−r τ−r τ it follows from this inequality and (4),that Z t y(t) ≤ e−σ(t−τ)y(τ) + (σ − δ0)e−σt eσsy(s)ds τ C Z τ C Z t + b e−σ(t−τ) kw(s)k2ds + b e−σt eσsk∇w(s)k2ds ε1 τ−r ε1λ1 τ Z t 2 2 1 −σt σs 2 k3 N+2 + e e kg(s)k ds + 1 + M N−2 . ε2 τ ε3σ For µ0 := δ0 − σ − Cb > 0 , one has ε1λ1 Z t k∇w(t)k2 + kw(t)k2 + µ0e−σt eσsk∇w(s)k2ds τ C Z τ ≤ e−σ(t−τ) k∇w(τ)k2 + kw(τ)k2 + b kw(s)k2ds ε1 τ−r Z t 2 2 1 −σt σs 2 k3 N+2 + e e kg(s)k ds + 1 + M N−2 . (52) ε2 τ ε3σ By this last estimate, we have k∇w(t)k2 + kw(t)k2 C Z τ ≤ e−σ(t−τ) k∇w(τ)k2 + kw(τ)k2 + b kw(s)k2ds ε1 τ−r Z t 2 2 1 −σt σs 2 k3 N+2 + e e kg(s)k ds + 1 + M N−2 , (53) ε2 τ ε3σ and Z t µ0e−σt eσsk∇w(s)k2ds τ C Z τ ≤ e−σ(t−τ) k∇w(τ)k2 + kw(τ)k2 + b kw(s)k2ds ε1 τ−r EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 23

Z t 2 2 1 −σt σs 2 k3 N+2 + e e kg(s)k ds + 1 + M N−2 . (54) ε2 τ ε3σ For t − r ≥ τ, we have Z t Z t eσsk∇w(s)k2ds ≥ eσsk∇w(s)k2ds τ t−r Z t ≥ eσ(t−r) k∇w(s)k2ds . t−r So, by this inequality, (54) becomes Z t k∇w(s)k2ds t−r C Z τ ≤ µ0−1e−σ(t−τ−r) k∇w(τ)k2 + kw(τ)k2 + b kw(s)k2ds ε1 τ−r 0−1 Z t 2 2 µ −σ(t−r) σs 2 0−1 k3 N+2 σr + e e kg(s)k ds + µ 1 + M N−2 e . (55) ε2 τ ε3σ We add (53) and (55), and using (19), we ﬁnd Z t Z t k∇w(t)k2 + k∇w(s)k2ds ≤ k∇w(t)k2 + kw(t)k2 + k∇w(s)k2ds t−r t−r C Z τ ≤ e−σ(t−τ) k∇w(τ)k2 + kw(τ)k2 + b kw(s)k2ds 1 + µ0−1eσr ε1 τ−r Z t 2 2 0−1 σr 1 −σt σs 2 k3 N+2 + 1 + µ e e e kg(s)k ds + 1 + M N−2 . (56) ε2 −∞ ε3σ

Then, (56) shows that for all ε > 0 , τ ≤ τ0 and all (u(t), ut) ∈ U(t, τ)B(τ) , one has k(I − P )(u(t), u )k2 ≤ ε2 . n t X1 In what follows, we need the following proposition.

2 1 Proposition 3. Let v ∈ L (Ω) and λm be the greater eigenvalue of −∆ in H0 (Ω) . Then we have 2 2 2 2 k∆vk ≤ λmk∇vk ≤ λmkvk . (57) Proof of the Proposition. By the Parseval equality, one has

2 X 2 k∆vk = | h∆v, eii | . i≥1 Because of −∆ = ∇2 is selfadjoint, we have

2 X 2 k∆vk = | h∇v, ∇eii | . i≥1 √ √ Since ∇ei = λiei such that λi are the eigenvalues of ∇ , we ﬁnd

By the fact that λi ≤ λm →m→+∞ +∞ , we get 2 X 2 k∆vk ≤ λm | h∇v, eii | . i≥1 Hence 2 2 k∆vk ≤ λmk∇vk . Similarly, we prove 2 2 k∇vk ≤ λmkvk , and we deduce the desired result. 3. Finally, it suﬃces to consider the ordinary functional diﬀerential system (43). We check the equicontinuity property of the solutions {v(·)} in the space L2([t − −1 + r, t]; H (Ω)) . Then, for any t1 ∈ [t−r, t] , any l ∈ R with l < δ and for [t1, t1 +l] ⊂ [t − r, t] , we have kTlv(t1) − v(t1)k = kv(t1 + l) − (t1)k Z l dv(t1 + s) ≤ ds 0 dt1 Z l Z l d ≤ ∆ v(t1 + s) ds + k∆v(t1 + s)kds 0 dt1 0 Z l Z l Z l

+ kf0(v)kds + kb(t1 + s, vt1+s)kds + kg(t1 + s)kds , 0 0 0 and so Z l Z l 2 d kv(t1 + l) − v(t1)k ≤ ∆ v(t1 + s) ds + k∆v(t1 + s)kds 0 dt1 0 Z l Z l

+ kf0(v)kds + kb(t1 + s, vt1+s)kds 0 0 Z l 2 + kg(t1 + s)kds . 0 Using the convexity property three times, one ﬁnds 2 Z l Z l ! 2 d kv(t1 + l) − v(t1)k ≤ 2 ∆ v(t1 + s) ds + k∆v(t1 + s)kds 0 dt1 0 Z l Z l

+ 2 kf0(v)kds + kb(t1 + s, vt1+s)kds 0 0 Z l 2 + kg(t1 + s)kds 0 2 2 Z l ! Z l ! d ≤ 4 ∆ v(t1 + s) ds + 4 k∆v(t1 + s)kds 0 dt1 0 2 Z l Z l !

+ 4 kf0(v)kds + kb(t1 + s, vt1+s)kds 0 0 2 Z l ! + 4 kg(t1 + s)kds 0 EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 25

2 2 Z l ! Z l ! d ≤ 4 ∆ v(t1 + s) ds + 4 k∆v(t1 + s)kds 0 dt1 0 2 2 Z l ! Z l !

+ 8 kf0(v)kds + 8 kb(t1 + s, vt1+s)kds 0 0 2 Z l ! + 4 kg(t1 + s)kds . 0 By the H¨olderinequality, we obtain

2 kv(t1 + l) − v(t1)k Z l 2 Z l d 2 ≤ 4l ∆ v(t1 + s) ds + 4l k∆v(t1 + s)k ds 0 dt1 0 Z l Z l Z l 2 2 2 + 8l kf0(v)k ds + 8l kb(t1 + s, vt1+s)k ds + 4l kg(t1 + s)k ds 0 0 0 Z l 2 Z l d 2 ≤ 8l ∆ v(t1 + s) ds + k∆v(t1 + s)k ds 0 dt1 0 Z l Z l Z l 2 2 2 + kf0(v)k ds + kb(t1 + s, vt1+s)k ds + kg(t1 + s)k ds . 0 0 0 Applying the fact that L2(Ω) ⊂ H−1(Ω) with continuous injection on the term in the left hand side of the previous inequality, one has

2 2 kv(t1 + l) − v(t1)kH−1(Ω) ≤ k4kv(t1 + l) − v(t1)k Z l 2 Z l d 2 ≤ 8lk4 ∆ v(t1 + s) ds + k∆v(t1 + s)k ds 0 dt1 0 Z l Z l Z l 2 2 2 + kf0(v)k ds + kb(t1 + s, vt1+s)k ds + kg(t1 + s)k ds . 0 0 0 Integrating this last estimate over [t − r, t], we ﬁnd Z t 2 kv(t1 + l) − v(t1)kH−1(Ω)dt1 t−r Z t Z l 2 Z l d 2 ≤ 8lk4 ∆ v(t1 + s) ds + k∆v(t1 + s)k ds t−r 0 dt1 0 Z l Z l Z l 2 2 2 + kf0(v)k ds + kb(t1 + s, vt1+s)k ds + kg(t1 + s)k ds dt1 0 0 0 Z l Z t 2 Z l Z t d 2 ≤ 8lk4 ∆ v(t1 + s) dt1ds + k∆v(t1 + s)k dt1ds 0 t−r dt1 0 t−r Z l Z t Z l Z t 2 2 + kf0(v)k dt1ds + kb(t1 + s, vt1+s)k dt1ds 0 t−r 0 t−r Z l Z t 2 + kg(t1 + s)k dt1ds . (58) 0 t−r Next, we will estimate the ﬁve terms on the right hand of the equation. 26 MUSTAPHA YEBDRI

(i) By (57), we have Z t Z t 2 2 k∆v(t1 + s)k dt1 ≤ λm k∇v(t1 + s)k dt1 . t−r t−r From (20), one has

Z t 2N Z t 2 0 N−2 −σ(t1+s−τ) k∇v(t1 + s)k dt1 ≤ ck(u , ϕ)kX e dt1 t−r t−r t t1+s Z Z 0 −σ(t1+s) σs 0 2 0 + c 1 + e e kg(s )k ds dt1 t−r −∞ c 2N ≤ k(u0, ϕ)k N−2 e−σ(t−r+s−τ) − e−σ(t+s−τ) + cr σ X t c Z 0 + e−σ(t−r+s) − e−σ(t+s) eσs kg(s0)k2ds0 σ −∞ c 2N ≤ k(u0, ϕ)k N−2 e−σ(t−r+s−τ) σ X t c Z 0 + e−σ(t−r+s) eσs kg(s0)k2ds0 + cr . (59) σ −∞ Integrating over [0, l] , we obtain Z l Z t 2 k∇v(t1 + s)k dt1ds 0 t−r c 2N ≤ k(u0, ϕ)k N−2 e−σ(t−r−τ)(1 − e−σl) σ2 X Z t c −σ(t−r) −σl σs 2 + 2 e (1 − e ) e kg(s)k ds + crl . (60) σ −∞ By (57) and (60), we have Z l Z t 2 k∆v(t1 + s)k dt1ds 0 t−r c 2N ≤ λ k(u0, ϕ)k N−2 e−σ(t−r−τ)(1 − e−σl) m σ2 X Z t c −σ(t−r) −σl σs 2 + λm 2 e (1 − e ) e kg(s)k ds + λmcrl σ −∞ → 0 as l → 0 . (61) (ii) From (bII) and (bIV), we obtain Z t Z t 2 2 kb(t1 + s, vt1+s)k dt1 ≤ Cb kv(t1 + s)k dt1 . t−r t−2r Using the (4), one has Z t Z t 2 −1 2 kb(t1 + s, vt1+s)k dt1 ≤ Cbλ1 k∇v(t1 + s)k dt1 . t−r t−2r By (20), one gets

Z t 2N Z t 2 −1 0 N−2 −σ(t1+s−τ) kb(t1 + s, vt1+s)k dt1 ≤ Cbλ1 ck(u , ϕ)kX e dt1 t−r t−2r EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 27

t t t1+s Z Z Z 0 −1 −1 −σ(t1+s) σs 0 2 0 + Cbλ1 c dt1 + Cbλ1 c e e kg(s )k ds dt1 t−2r t−2r −∞ c 2N ≤ C λ−1 k(u0, ϕ)k N−2 e−σ(t−2r+s−τ) − e−σ(t+s−τ) b 1 σ X Z t −1 −1 c −σ(t−2r+s) −σ(t+s) σs0 0 2 0 + 2Cbλ1 cr + Cbλ1 e − e e kg(s )k ds σ −∞ c 2N ≤ C λ−1 k(u0, ϕ)k N−2 e−σ(t−2r+s−τ) b 1 σ X Z t −1 −1 c −σ(t−2r+s) σs0 0 2 0 + 2Cbλ1 cr + Cbλ1 e e kg(s )k ds . (62) σ −∞ Integrating over [0, l] , we get

Z l Z t 2N Z l 2 −1 c 0 N−2 −σ(t−2r+s−τ) kb(t1 + s, vt1+s)k dt1ds ≤ Cbλ1 k(u , ϕ)kX e ds 0 t−r σ 0 Z l Z t Z l −1 c −σ(t−2r+s) σs0 0 2 0 −1 + Cbλ1 e e kg(s )k ds ds + 2Cbλ1 cr ds σ 0 −∞ 0 c 2N ≤ C λ−1 k(u0, ϕ)k N−2 e−σ(t−2r−τ) 1 − e−σl b 1 σ2 X Z t −1 c −σ(t−2r) −σl σs0 0 2 0 −1 + Cbλ1 2 e 1 − e e kg(s )k ds + 2Cbλ1 crl σ −∞ → 0 when l → 0 . (63)

2 2 (iii) Similarly, since g ∈ Lloc(R; L (Ω)) and t − r ≤ t1 + s ≤ t , we get Z l Z t Z l Z t 2 0 2 0 kg(t1 + s)k dt1ds ≤ kg(s )k ds ds 0 t−r 0 t−r Z t ≤ l kg(s0)k2ds0 t−r 2 ≤ lkgkL2([t−r,t];L2(Ω)) → 0 as l → 0 . (64) (iv) Now, it is clear to see from (41) that Z 2 2 α 2 kf0(v)k ≤ k (1 + |v| ) dx . Ω 4N By the convexity of the power,(4) and the fact that 2α < N−2 , one has Z 2 2 2α kf0(v)k ≤ 2k (1 + |v| )dx Ω 2 2 2α ≤ 2k |Ω| + 2k kv(t1 + s)k 4N 2 2 −1 N−2 ≤ 2k |Ω| + 2k µλ˜ 1 k∇v(t1 + s)k . (65) Where, by (20), we have

4N 2N N−2 −σ(t1+s−τ) 0 N−2 k∇v(t1 + s)k ≤ ce k(u , ϕ)kX

2N t1+s Z 0 N−2 + c 1 + e−σ(t1+s) eσ(s )kg(s0)k2ds0 . −∞ 28 MUSTAPHA YEBDRI

Similarly, using the convexity of the power tow times, one obtains

2N N−2 4N 2N 2N N−2 N−2 −1 −σ(t1+s−τ) 0 N−2 k∇v(t1 + s)k ≤ 2 ce k(u , ϕ)kX

2N Z t1+s N−2 2N −1 2N −σ(t +s) σs0 0 2 0 + 2 N−2 c N−2 1 + e 1 e kg(s )k ds −∞ 2 N+2 2N 2N ( 2N ) 2N+4 2N N−2 N−2 −σ(t1+s−τ) N−2 0 N−2 N−2 N−2 ≤ 2 c e k(u , ϕ)kX + 2 c 2N Z t1+s N−2 2N+4 2N −σ(t +s) 2N σs0 0 2 0 + 2 N−2 c N−2 e 1 N−2 e kg(s )k ds . (66) −∞ Hence by (65) and (66), one gets

t t t Z Z Z 4N 2 2 2 −1 N−2 kf0(v)k dt1 ≤ 2k |Ω| 1dt1 + 2k µλ˜ 1 k∇v(t1 + s)k dt1 t−r t−r t−r 2 2 N+2 2N 2N 2N 2N k −1 −σ(t+s−τ) −σ(t−r+s−τ) 0 ( N−2 ) ≤ 2 µλ˜ 2 N−2 c N−2 (−e N−2 + e N−2 )k(u , ϕ)k σ 1 X 2 k −1 N+2 2N −σ(t−r+s) 2N + 2 µλ˜ 2 N−2 c N−2 (e N−2 σ 1 2N Z t N−2 −σ(t+s) 2N σs0 0 2 0 − e N−2 ) e kg(s )k ds −∞ 3N+2 2N 2 N−2 N−2 2 −1 + 2k |Ω|r + 2 c k µλ˜ 1 r 2 2 N+2 2N 2N 2N k −1 −σ(t−r+s−τ) 0 ( N−2 ) ≤ 2 µλ˜ 2 N−2 c N−2 e N−2 k(u , ϕ)k σ 1 X 2N 2 t N−2 k N+2 2N 2N Z 0 −1 N−2 N−2 −σ(t−r+s) N−2 σs 0 2 0 + 2 µλ˜ 1 2 c e e kg(s )k ds σ −∞ 3N+2 2N 2 N−2 N−2 2 −1 + 2k |Ω|r + 2 c k µλ˜ 1 r . We integrate from 0 to l to get Z l Z t 2 kf0(v)k dt1ds 0 t−r

2 2N 2 Z l k N+2 2N ( ) 2N −1 N−2 N−2 0 N−2 −σ(t−r+s−τ) N−2 ≤ 2 µλ˜ 1 2 c k(u , ϕ)kX e ds σ 0 2N 2 t N−2 l k N+2 2N Z 0 Z 2N −1 N−2 N−2 σs 0 2 0 −σ(t−r+s) N−2 + 2 µλ˜ 1 2 c e kg(s )k ds e ds σ −∞ 0 l Z 3N+2 2N 2 N−2 N−2 2 −1 + 2k |Ω|r + 2 c k µλ˜ 1 r ds 0 Then, we ﬁnd Z l Z t 2 kf0(v)k dt1ds 0 t−r 2 2 N+2 2N 2N 2N 2N k −1 0 ( N−2 ) −σ(t−r−τ) −σl ≤ 2 µλ˜ 2 N−2 c N−2 k(u , ϕ)k e N−2 (1 − e N−2 ) σ2 1 X EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 29

2N 2 t N−2 k N+2 2N Z 0 2N 2N −1 N−2 N−2 σs 0 2 0 −σ(t−r) N−2 −σl N−2 + 2 2 µλ˜ 1 2 c e kg(s )k ds e (1 − e ) σ −∞ 3N+2 2N 2 N−2 N−2 2 −1 + 2k |Ω|rl + 2 c k µλ˜ 1 rl → 0 as l → 0 . (67) 2 R l R t d (v) Now, it remains to estimate ∆ v(t1 + s) dt1ds . To this end, we 0 t−r dt1 take the inner product in L2(Ω) of (43) with −∆ ∂ v , we ﬁnd ∂t1 2 2 d d 1 d 2 ∇ v(t1 + s) + ∆ v(t1 + s) + k∆v(t1 + s)k dt1 dt1 2 dt1 Z ∂ Z ∂ ≤ f0(v)∆ v + b(t1 + s, vt1+s) −∆ v Ω ∂t1 Ω ∂t1 Z ∂ + g(t1 + s) −∆ v . (68) Ω ∂t1 Although, we have Z Z ∂ 0 ∂ f0(v)∆ v = − f0(v)∇v∇ v . Ω ∂t1 Ω ∂t1 By (40), it follows that Z Z ∂ ∂ f0(v)∆ v ≤ c3 |∇v| ∇ v Ω ∂t1 Ω ∂t1

∂ ≤ c3k∇vk ∇ v . ∂t1 Using the Young inequality, one has Z 2 2 ∂ c3 2 1 d f0(v)∆ v ≤ k∇v(t1 + s)k + ∇ v(t1 + s) . (69) Ω ∂t1 2 2 dt1 By (69) and (68), one ﬁnds 2 2 d d 1 d 2 ∇ v(t1 + s) + ∆ v(t1 + s) + k∆v(t1 + s)k dt1 dt1 2 dt1 2 2 Z c3 2 1 d ∂ ≤ k∇v(t1 + s)k + ∇ v(t1 + s) + b(t1 + s, vt1+s) −∆ v 2 2 dt1 Ω ∂t1 Z ∂ + g(t1 + s) −∆ v . Ω ∂t1 We use the Cauchy inequality, we obtain 2 2 d d 1 d 2 ∇ v(t1 + s) + ∆ v(t1 + s) + k∆v(t1 + s)k dt1 dt1 2 dt1 2 2 c3 2 1 d 1 2 ≤ k∇v(t1 + s)k + ∇ v(t1 + s) + kb(t1 + s, vt1+s)k 2 2 dt1 2ν1 2 2 ν1 d 1 2 ν2 d + ∆ v(t1 + s) + kg(t1 + s)k + ∆ v(t1 + s) . 2 dt1 2ν2 2 dt1 Hence, one has 2 d ∆ v(t1 + s) dt1 30 MUSTAPHA YEBDRI

2 2 1 d d 1 d 2 ≤ ∇ v(t1 + s) + ∆ v(t1 + s) + k∆v(t1 + s)k 2 dt1 dt1 2 dt1 2 2 c3 2 1 2 ν1 + ν2 d ≤ k∇v(t1 + s)k + kb(t1 + s, vt1+s)k + ∆ v(t1 + s) 2 2ν1 2 dt1

1 2 + kg(t1 + s)k . 2ν2 Therefore, one gets

2 2 ν1 + ν2 d c3 2 1 2 1 − ∆ v(t1 + s) ≤ k∇v(t1 + s)k + kb(t1 + s, vt1+s)k 2 dt1 2 2ν1

1 2 + kg(t1 + s)k . 2ν2

ν1+ν2 For ν1, ν2 small enough, we have ν3 := 1 − 2 > 0 . So we can write

2 2 d c3 2 1 2 ∆ v(t1 + s) ≤ k∇v(t1 + s)k + kb(t1 + s, vt1+s)k dt1 2ν3 2ν1ν3

1 2 + kg(t1 + s)k . 2ν2ν3 We integrate this last one over [t − r, t] , one has

Z t 2 2 Z t d c3 2 ∆ v(t1 + s) dt1 ≤ k∇v(t1 + s)k dt1 t−r dt1 2ν3 t−r Z t Z t 1 2 1 2 + kb(t1 + s, vt1+s)k dt1 + kg(t1 + s)k dt1 . 2ν1ν3 t−r 2ν2ν3 t−r Integrating it over [0, l] , one obtains

Z l Z t 2 2 Z l Z t d c3 2 ∆ v(t1 + s) dt1ds ≤ k∇v(t1 + s)k dt1ds 0 t−r dt1 2ν3 0 t−r Z l Z t Z l Z t 1 2 1 2 + kb(t1 + s, vt1+s)k dt1ds + kg(t1 + s)k dt1ds . 2ν1ν3 0 t−r 2ν2ν3 0 t−r By (60), (63) and (64), it follows that Z l Z t 2 d ∆ v(t1 + s) dt1ds 0 t−r dt1 2 2N c3 c 0 N−2 −σ(t−r−τ) −σl ≤ 2 k(u , ϕ)kX e (1 − e ) 2ν3 σ 2 Z t 2 c3 c −σ(t−r) −σl σs 2 c3 + 2 e (1 − e ) e kg(s)k ds + crl 2ν3 σ −∞ 2ν3 2N 1 −1 c 0 N−2 −σ(t−2r−τ) −σl + Cbλ1 2 k(u , ϕ)kX e 1 − e 2ν1ν3 σ Z t 1 −1 c −σ(t−2r) −σl σs 2 1 −1 + Cbλ1 2 e 1 − e e kg(s)k ds + Cbλ1 crl 2ν1ν3 σ −∞ ν1ν3 1 2 + lkgkL2([t−r,t];L2(Ω)) . 2ν2ν3 EXISTENCE OF D−PULLBACK ATTRACTOR FOR AN IDDS 31

Thus, we get Z l Z t 2 d ∆ v(t1 + s) dt1ds 0 t−r dt1 2 2N c3 1 −1 σr c 0 N−2 −σ(t−r−τ) −σl ≤ + Cbλ1 e 2 k(u , ϕ)kX e (1 − e ) 2ν3 2ν1ν3 σ 2 Z t c3 1 −1 σr c −σ(t−r) −σl σs 2 + + Cbλ1 e 2 e (1 − e ) e kg(s)k ds 2ν3 2ν1ν3 σ −∞ 2 c3 1 −1 1 2 + crl + Cbλ1 crl + lkgkL2([t−r,t];L2(Ω)) → 0 as l → 0 . (70) 2ν3 ν1ν3 2ν2ν3 Comprehensively, from (58), (61), (63), (64), (67) and (70), we have Z t 2 kv(t1 + l) − v(t1)kH−1(Ω)dt1 → 0 when l → 0 , t−r which implies the needed equicontinuity. This shows that the condition (3) in Proposition2 holds true, and thus the process on X is pullback w-D-limit compact. Then from Lemma 4.3 and Theorem 4.5, we conclude the existence of pullback D-attractor which completes the proof of the theorem.

Acknowledgments. The author thanks the referee for the comments and sugges- tions that improved the contents and readability of the article.

REFERENCES

[1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [2] J. Barrow-Green, Poincaréand the three-body problem, History of Mathematics, vol. 11. Amer. Math.Soc., Providence, RI, 1997. [3] P. Bergé,Y. Pomeau and C. Vidal, L’ordre Dans le Chaos, Hermann, Paris, 1984. [4] J. E. Bilotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 (1971), 1082–1088. [5] E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive nonautonomous dynamical systems and their relations, J. D ifferential Equations, 262 (2017), 3524–3550. [6] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J.Differential Equations, 257 (2014), 490–522. [7] H. Brezis, Analyse Fonctionnelle, Théorieet Applications, Masson, Paris, 1983. [8] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems AKTA, Kharkiv, 1999. [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [10] J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations Springer-Verlag, 1993. [11] H. Haraga and M. Yebdri, Pullback attractors for class of semilinear nonclassical diffusion equation with delay, Applied Mathematics and Nonlinear Sciences, 3 (2018), 127–150. [12] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Cambridge 1991. [13] Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and ap- plication to the nonautonomous reaction-diffusion equations, J. Applied Mathematics and Computation, 190 (2007), 1020–1029. [14] J.-L. Lions, Quelques Méthodes de Résolutiondes Problèmes aux Limites Non Linéaires Dunod, Paris 1969. 32 MUSTAPHA YEBDRI

[15] X. Liu and Y. Wang, Pullback attractors for nonautonomous 2D-Navier-Stokes models with variable elays, J. Abstract and Appl. Anal., (2013), Art. ID 425031, 10 pp. [16] J. C. Robinson, Infinite-Dimensional Dynamical Systems Cambridge University Press, Cam- bridge, 2001. [17] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (2nd edition), Springer-Verlag, New York, 1997. [18] C. The Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, J. Nonlinear Anal., 73 (2010), 399–412. [19] Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, J. Discrete and Continuous Dynamical Systems, 34 (2014), 4343–4370. [20] J. Wu, Theory and Applications of Partiali Functional Differential Equations Springer- Verlag, New York, 1996. Received April 2020; revised December 2020.

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