ATTRACTORS FOR A CLASS OF DELAYED REACTION-DIFFUSION EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS

. In this paper we study the asymptotic behavior of solutions for a class of nonautonomous reaction-diﬀusion equations with dynamic bound- ary conditions possessing ﬁnite delay. Under the polynomial conditions of reaction term, suitable conditions of delay terms and a minimal conditions of time-dependent force functions, we ﬁrst prove the existence and uniqueness of solutions by using the Galerkin method. Then, we ensure the existence of pullback attractors for the associated process to the problem by proving some uniform estimates and asymptotic compactness properties (via an energy method). With an additional condition of time-dependent force functions, we prove that the boundedness of pullback attractors in smoother spaces.

1. Introduction. Partial differential equations with dynamic boundary conditions arise for example in hydrodynamics and the heat transfer theory. For instance, they allow to model heat flow inside the considered domain subject to nonlinear heating or cooling at the boundary, or heat transfer in a solid in contact with a moving fluid, in thermoelasticity, diffusion phenomena, heat transfer in two mediums, etc. [2,8]. The long-time behavior of solutions for reaction-diffusion equations with dynamic boundary conditions has been studied extensively (see [1,3,7,15,16,17,18]).
Delayed differential equations arise in many realistic models of problems in science and engineering where there is a time lag or after-effect. In particular, the parabolic case represents some issues in mathematical biology and the time lags are often seen as maturation time for population dynamics. Let us introduce some relevant literatures in [9,10,13]. It is naturally to study time-dependent partial differential equations with dynamic boundary conditions concerning with delays, especially the reaction-diffusion equations with dynamic boundary conditions involving delays.
The long-time behavior of solutions of the delayed reaction-diffusion equation with Dirichlet boundary condition were studied in [5,11,12]. To the best of our knowledge, there is only paper [14] considered the long-time behavior of solutions for the delayed equations with dynamic boundary conditions. More precisely, the authors studied the existence of pullback attractors for a p-Laplacian nonautonomous problem with dynamic boundary conditions and infinite delay. The p-Laplace operator becomes the Laplace operator when p = 2, but in the problem considered in [14], the reaction terms disappeared. So, up to now, there is no result concerning with the delayed reaction-diffusion equations with dynamic boundary conditions. In this paper, we study the long-time behavior by analyzing the existence of pullback attractors of a class of reaction-diffusion equations with dynamic boundary conditions and finite delay. We also mention here that, in [14], the authors considered the case of p-Laplace operator with infinite delay. Thus, there are some differences in the proofs of the existence of solutions and pullback attractor, especially in the proof of the asymptotic compactness since the phase space is needed to be considered is different to the phase space in this paper.
Let Ω ⊂ R d , d ∈ N + , be an open bounded domain with smooth boundary Γ = ∂Ω. We consider the following reaction-diffusion equation equations with dynamic boundary conditions and finite delay: in Ω × (τ, ∞), ∂ t u + ∂ n u + κu + f Γ (u) = g Γ (t, u t ) + ρ Γ (x, t) on Γ × (τ, ∞), where τ ∈ R, n denotes the outward normal vector at Γ, κ > 0, φ ∈ C([−h, 0]; L 2 (Ω)), φ Γ ∈ C([−h, 0]; L 2 (Γ)) are the initial datum, h > 0 being the length of the delay effect, and where for each t ≥ τ , we denote by u t (θ) = u(t + θ), θ ∈ [−h, 0]. We consider (1) with the following conditions: (h1) f, f Γ : R → R are continuous functions satisfying and where c 0 , c 1 , c 2 and are positive constants. (h2) The delay functions (g, g Γ ) : ii) ∀t ∈ R, g(t, 0) = 0, g Γ (t, 0) = 0, iii) ∃L g > 0 such that ∀t ∈ R, ∀(ξ, ξ), (η, η) ∈ C([−h, 0]; L 2 (Ω) × L 2 (Γ)), Here, we follow the conditions of delay functions which were studied in [5] for delayed reaction-diffusion equations with homogeneous Dirichlet boundary conditions. From (2) and (4) then there exist two positive constants c 3 , c 4 such that We denote the primitive functions of f and f Γ by F (s) = s 0 f (r)dr and F Γ (s) = s 0 f Γ (r)dr. From conditions (2) and (4), there exist positive constantsc 0 ,c 1 andc 2 such thatc The rests of paper is organized as follows. In the next section we present some preliminaries. In Section 3, with above condition and a minimal conditions of timedependent force functions, we prove the existence and uniqueness of solution by using the Galerkin method. In the last section, after recall some abstract results of pullback attractors theory [6], we analyze conditions in order to obtain two different families of minimal pullback attractors, namely, those of fixed bounded sets but also for a class of time-dependent families (universe) given by a tempered condition, for the natural (nonautonomous) dynamical system associated to the problem through the previous result. Finally, we also prove the minimal pullback attractor is bounded in some smoother spaces under an additional condition of time-dependent force functions.
From (11) and (12) we have the following estimate where , and where We consider the operator A : V → V * given by Then from (13) the operator A is coercive and then by the Lax-Milgram theorem, A has a bounded inverse A −1 : V * → V with its restriction to H is a compact operator since the compactness of the embbeding V ⊂ H. Moreover, from (14) then A : D(A) → H is positive. Hence, since A is self-adjoint, positive operator with compact resolvent, there exists an orthornomal basis {(w j , γ 0 (w j ))} ∞ j=1 ⊂ D(A) in H, consisting of eigenfunctions of A with corresponding eigenvalues {λ j } ∞ j=1 ⊂ R + , forms an increasing sequence converging to infinity. Moreover {(w j , γ 0 (w j ))} ∞ j=1 is a orthogonal basis in V .
3. Existence and uniqueness of weak solutions.
We prove the existence and uniqueness of weak solutions to (1) by using the Galerkin approximation method.
given. Then there exists a unique weak solution (u, u Γ ) to (1) and the solution depends continuously on the initial data. Proof.
Step 1. Existence. We first prove the existence of weak solution to (1) by using the Galerkin method.
Step 1.1. The Galerkin approximation. Let P m be the orthonormal projector from H to span{(w 1 , γ 0 (w 1 )), . . . , (w m , γ 0 (w m ))}, where {(w j , γ 0 (w j ))} ∞ j=1 is the basis of all the eigenfunctions of the operator A which is orthonormal in H and orthogonal in V . We consider the approximation solution where χ mj (t) are required to satisfy the following differential equation system with finite delay: (15) It is well-known that the above finite-dimensional delayed system is well-posed [10], at least locally. Hence, we conclude that the approximation solution u m (t) to (15) exists unique locally on [τ, T m ) with τ ≤ T m ≤ T . Next, we will obtain a priori estimates and ensure that the solutions u m exists in the whole interval [τ − h, T ].
Step 1.2. A priori estimates. Multiplying the first equation in (15) by χ mj (t) and summing up from j = 1 to m, we obtain Using the conditions (h1)-(h2), the Cauchy inequality and using (13), we have Thus, Using (13) then we get from (16) that for any t ≥ τ , In particular, And then Using the Gronwall inequality, we deduce that From this and (17) we conclude that Using (7)- (8) and (19) then where p = p p−1 , q = q q−1 . From these bounds we get from the first and the second equations in (1) that Step 1.3. Passing to the limits.
From the bounds (18) and (21)-(22), it follows from the Aubin-Lions compact lemma that So from the continuity of f, f Γ we conclude that And then we have that χ = f (u), ζ = f Γ (γ 0 (u)).
We now have to passing to the limit of the nonlinear function related to delay term. To do this, we show that We have We first see that Thus, to obtain (23), we need show that Indeed, let (u n , γ 0 (u n )) and (u m , γ 0 (u m )) satisfy (15) then w = u n − u m satisfies Using conditions (3), (5) and (6) we deduce from (25) that Thus, By the Gronwall inequality we obtain that ;H) . This shows that {(u n − u m , γ 0 (u n − u m ))} is a Cauchy sequence in C([τ, T ]; H). So, we get (24). And consequence we get (23) and . Therefore, we can pass to the limits to get that (u, γ 0 (u)) is the solution of (1).

Some concepts of pullback attractors.
For the convenience of the reader, we recall in this section some concepts and results on the theory of pullback Dattractors (see [6]), which will be used in the paper. Let X be a metric space with metric d X . A process in X is a mapping U (t, τ ) : X → X such that U (τ, τ ) = Id and U (t, s)U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ, τ ∈ R. A process {U (t, τ )} t≥τ is said to be continuous if for any τ ≤ t, the map U (t, τ ) : X → X is continuous. The process {U (t, τ } t≥τ is said to be closed if for any τ ≤ t, any sequence {x m } ⊂ X with x m → x ∈ X and U (t, τ )x m → y ∈ X, then U (t, τ )x = y. We note that if a process is continuous then it is closed.
Denote by P(X) the set of all nonempty subsets of X and consider a family of nonempty setsD 0 = {D 0 (t) : t ∈ R} ⊂ P(X).
Definition 4.1. We say that a process {U (t, τ )} t≥τ is pullbackD 0 -asymptotically compact if for any t ∈ R, any any sequence {τ m } with τ m ≤ t for all m satisfying τ m → −∞, any sequence x m ∈ D(τ m ) for all m, the sequence {U (t, τ m )x m } is relatively compact in X.
Let be given D a nonempty class of families parameterized in timeD = {D(t) : t ∈ R} ⊂ P(X). The class D will be called a universe in P(X). Denote Given two subset of X, O 1 , O 2 , we denote by dist X (O 1 , O 2 ) the Hausdorff semidistance in X between them.

Definition 4.2.
A process {U (t, τ )} t≥τ on X is called pullback D-asymptotically compact if it is pullbackD-asymptotically compact for anyD ∈ D.
It is said thatD 0 = {D 0 (t) : t ∈ R} ⊂ P(X) is pullback D-absorbing for the process {U (t, τ )} t≥τ in X if for any t ∈ R and anyD ∈ D, there exists The family A D is minimal in the sense that ifĈ = {C(t) : t ∈ R} ⊂ P(X) is a family of closed sets such that for anyD ∈ D, lim Remark 1. If A D ∈ D then it is the unique family of closed subsets in D that satisfies (2)-(3).
A sufficient condition for A D ∈ D is to have thatD 0 ∈ D, the set D 0 (t) is closed for all t ∈ R, and D is inclusion closed (i.e., ifD ∈ D, andD = {D (t) : t ∈ R} ⊂ P(X) with D (t) ⊂ D(t) for all t, thenD ∈ D).
Let D F (X) the universe of fixed nonempty bounded subsets of X, i.e., the class of all familiesD = {D(t) = D : t ∈ R} with D a fixed nonempty bounded subset of X.

Corollary 1. Under the assumptions of Theorem 4.3, if D contains D F (X) then both attractors, A D F (X) and A D exist and
Moreover, if for some T ∈ R, the set t≤T D 0 (t) is bounded subset of X, then  We denote B X (R) is the closed ball in X centered at 0 with radius R.