PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY NON-AUTONOMOUS PROBLEM WITH INFINITE DELAY

. In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and inﬁnite delay. We en- sure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.


1.
Introduction. Let τ ∈ R and Ω ⊂ R N be a bounded domain with smooth boundary Γ = ∂Ω and N ≥ 3, consider the following dynamical boundary conditions problem with infinite delay    u t − ∆ p u + |u| p−2 u = f 1 (t, u t ) + g 1 (t, x), (t, x) ∈ (τ, +∞) × Ω, u t + |∇u| p−2 ∂− → n u = f 2 (t, u t ) + g 2 (t, x), (t, x) ∈ (τ, +∞) × Γ, u(τ + s, x) = Ψ(s, x), s ∈ (−∞, 0], x ∈ Ω (P ) where − → n is the outer normal to Γ, p ∈ [2, +∞) and ∆ p denotes the p-Laplacian operator, defined by ∆ p u = div(|∇u| p−2 ∇u). The external forces g i , i = 1, 2, satisfy assumptions that will be stated later, Ψ is a given function defined in the interval (−∞, 0] and the external force field f i containing some hereditary characteristic denoted by u t , which is a function defined on (−∞, 0) by the relation u t (s) = u(t+s), s ∈ (−∞, 0). The interest for problems with dynamic boundary conditions has been growing over the last forty years, see [5,11,13]. Motivated by mathematicians' interests and physical applications, the authors of [7] and [8] studied an autonomous version of Problem (P ). After that, some works emerged of this problem, with a nonautonomous term just in perturbations g i can be found in [12,20] and [21], where the authors have established the existence of a uniform attractor and pullback 510 RODRIGO SAMPROGNA AND TOMÁS CARABALLO attractor for the problems, respectively. In [18] the authors considered a nonautonomous term in perturbations f i and ensured the existence of solution as well as the existence of D-pullback attractor for the generalized process associated with a similar problem to (P ) without uniqueness of solution.
The delay terms appear naturally in many applications as velocity field in wind tunnel and population growth, e.g., [14]. The study of the asymptotic behaviour of problems with finite delay with uniqueness or in multivalued contexts can be found in [3], a version with infinite delays can be found in [2], both works consider autonomous and non-autonomous problems. In the work [19] the authors developed a theory of pullback attractors for multivalued process associated with infinite delay problems and they established conditions to guarantee the existence of an invariant pullback attractor for this multivalued process. Our work in this paper will be based on these results. Another thing that motivates us is that there are only a few delay problems related to operator ∆ p which is a very good example of a nonlinear maximal monotone operator.
We organize this work as follows. In the next section, we recall some notations, definitions and properties of suitable spaces for the study of Problem (P ). In Section 3 we present some definitions and a result that ensures the existence of the pullback attractor in multivalued context developed in [19]. In Section 4 we prove the existence of weak solution for Problem (P ). Finally, Section 5 is devoted to ensure the existence of pullback attractor for our problem.

2.
Preliminaries. In this section, following [6] we define the appropriate spaces to study Problem (P ).

PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH DELAY 511
where γ : W 1,p (Ω) → W 1− 1 p ,p (Γ) is the continuous trace operator. In V p , we can consider the usual norm U V p = u W 1,p (Ω) + γ(u) . The space V p is densely and compactly contained in the Hilbert space X 2 for 2 ≤ p < +∞, as can be seen in [7]. Note that we can identify u ∈ W 1,p (Ω) as a couple U = (u, γ(u)) ∈ V p . The continuity of γ ensures the equivalence between the norms of W 1,p (Ω) and V p . We can show that V p is a reflexive and separable space for 1 < p < ∞ . Furthermore, 3. Abstract results. In this section we present a summary of definitions and results from [19], where the authors developed a theory of invariant pullback attractors in a multivalued context. Let (X, ρ) be a complete metric space. For x ∈ X, A, B ⊂ X and ε > 0 we define Denote by P(X) the nonempty subsets of X.
Definition 3.2. Let {U (t, τ )} be a multivalued process on X. We say that {U (t, τ )} is 1. pullback dissipative, if there exists a family of bounded sets D = {D(t)} t∈R in X such that for any bounded set B ⊂ X and each t ∈ R, there exists a τ 0 = τ 0 (B, t) ∈ R such that The family of sets D is known as pullback absorbing family; 2. pullback asymptotically upper semicompact in X if for each fixed t ∈ R and B ⊂ X bounded, any sequence {τ n } with τ n → −∞, {x n } ⊂ B, and {y n } with y n ∈ U (t, τ n )x n , this last sequence {y n } is precompact in X.

Definition 3.3. A family of nonempty compact subsets
A is pullback attracting, i.e., for every bounded set B of X and any fixed Definition 3.4. Let {U (t, τ )} be a multivalued process on X. We say that U (t, τ ) is upper semicontinuous (or U.S.C.) in x for fixed t ≥ τ , τ ∈ R, if x n → x, then for any y n ∈ U (t, τ )x n , there exist a subsequence y n k ∈ U (t, τ )x n k and y ∈ U (t, τ )x such that y n k → y in X.

Existence of solution.
Let λ > 0 be fixed and H a Hilbert space. One possibility to deal with infinite delays is to consider the space: which is a Banach space with the norm This space was considered in [15,19], the properties of this space that will allow us to deal with infinite delays can be found in [9]. Later we will set a more appropriate λ to our particular problem.
(Ω) and L 2 = L 2 (Γ), and satisfies the following assumptions: See [17] for examples of functions with these properties. And for g i,s we have the following assumption: (G1) let g 1 ∈ L p loc (R; L p (Ω)), g 2 ∈ L p loc (R; L p (Γ)) where p denotes the conjugate exponent of p, i.e., 1 p + 1 p = 1. Remark 1. Let Ψ ∈ C λ (X 2 ), then notice that there exists ψ(s) ∈ L 2 (Ω) and φ(s) ∈ L 2 (Γ) for each s ∈ (−∞, 0] such that Ψ = (ψ, φ). Moreover, Then, from the continuity of trace, a.e. in (τ, T ), for each T > τ ; Before showing the existence of a weak solution to Problem (P ), we obtain a priori estimates for a weak solution in the space X 2 . and for all t ≥ τ , with C, C ε , C 1 ,C and Θ positive constants independent of τ and t.
Proof. Let U be a weak solution of Problem (P ). Take V = U in (2), and from Hölder's and Young's inequalities we have Then, as the norm of V p is equivalent to the norm of W 1,p (Ω), there is a constant M Ω such that Take ε > 0 such that and multiplying by 2, incorporating the constants, and integrating between τ to t Thus, from Lemma 2.1 of [18], (F2) and (F3), there are κ 1 , κ 2 > 0 and C κ1 , C κ2 > 0, such that andC := C ε (C κ1 + C κ2 ). From Remark 1 we have for t ≥ τ . Further and, note that ensuring estimate (4) for all t ≥ τ , with this estimate and (7) we can deduce estimate (3).
Proof. We will define some appropriate operators to reformulate expression (2) in order to have a simpler functional formulation of our problem, see [7] and [18] for examples of the same method. Then let, for U, V ∈ V p , the following operator For each U ∈ V p we have β p U := β p (U, ·) ∈ (V p ) * and the operator β p : V p → (V p ) * is a maximal monotone operator, see [18]. And we define in the usual way, see [18] for more details. In this way, finding a weak solution of Problem (P ) is equivalent to find a function U with regularities of weak solution definition, and satisfying the following functional equation in L p (0, T ; (V p ) * ), see Remark 4.6 in [18]. In order to find a weak solution to Problem (P ), we use the Faedo-Galerkin approximation. Since X 2 is separable and V p is dense in X 2 , there is a orthonormal basis of X 2 contained in V p . We denote such basis by {Φ n = (φ n , ψ n ) ∈ X 2 ; n ∈ N}.
Given Ψ ∈ C λ (X 2 ) and T > τ we want to find a solution U n = n i=1 d i (t)Φ i ∈ K n for an n−dimensional version of problem (8), which is equivalent to find a solution of the following system of ordinary differential equations for all 1 ≤ i ≤ n and a.e. in [τ, T ], where ·, · denote the dual product between (V p ) * and V p .
The above system of ordinary functional differential equations with infinite delay fulffils the conditions for existence and uniqueness of local solution established in Theorem 1.1 of [10]. A priori estimates ensure that solutions do exist for all time in [τ, T ].
In particular, the previous limit and estimate (3) imply the existence of another constant (relabelled the same) C = C(τ, T, R) such that Then, this guarantees that β p U n is bounded in L p (τ, T ; (V p ) * ), see [18] for more details. Hypotheses (F2), (F3), (9) and recalling that Therefore, the limits of β p U n and F(t, U t n ) ensure that there exists a constant (relabelled the same) C(τ, T, R) such that The limits in (10) and (12) ensure that there is a subsequence (which we relabel the same) {U n }, and an element U ∈ L ∞ (τ, T ; From compactness results, see Theorems 1.4 and 1.5 page 32 of [4], the sequences in fact have the following convergences Note that, P r n Ψ → Ψ in C λ (X 2 ), and thanks to the strong convergence in C([τ, T ]; X 2 ) yield that [15,16] and [17] for details about both convergences.
The above convergence and hypotheses (F2) and (F3) imply that , which together with convergences (13) and the theory of maximal monotone operators allow us to deduce that see [18] for details. Therefore, U is solution of the limit equation of (11) in the weak star topology of L p (τ, T ; (V p ) * ). This ensures that U is a weak solution of Problem (P ) in the interval (−∞, T ] with initial condition U τ = Ψ.

PULLBACK ATTRACTOR FOR A DYNAMIC BOUNDARY PROBLEM WITH DELAY 517
The existence of solution allows us define the multivalued process {U (t, τ )} on C λ (X 2 ) by Indeed, item (2) of Definition 3.1 follows from concatenation and translation of solutions, see [2] and [3] for details. Proof. Let τ ∈ R, {Ψ n } n∈N and Ψ such that Ψ n → Ψ in C λ (X 2 ), and let {Y n } n∈N such that Y · n ∈ U (·, τ )Ψ n . Given T > τ , observe that, as Ψ n → Ψ in C λ (X 2 ), given R > 0, except for a finite number of elements, we have that {Ψ n } ⊂ B C λ (X 2 ) (Ψ, R). Then, from Lemma 4.2 the sequence {Y n } is bounded in L ∞ (τ, T ; X 2 ) and L p (τ, T ; V p ).
Then, similarly to the proof of Theorem 4.3, we can ensure the existence of an element Y · ∈ U (·, τ )Ψ such that Y t n → Y t in C λ (X 2 ) for all t ≤ T . Therefore, as T > τ is arbitrary, it follows that {U (t, τ )} is uppersemicontinuous.

5.
Pullback attractor for problem (P ). In this section we develop some estimates to show that the multivalued process generated by solutions of Problem (P ) possesses a pullback absorbing family and it is pullback asymptotically upper semicompact. Therefore, we can ensure the existence of pullback attractor for the problem.
First we summary some aspects of the constants that appeared in the development of Lemma 4.2 and we will develop some technical property with these constants to make easy our study and understanding of the reader. Consider in this section p > 2.
Note that there is a constantM > 0 such that U X 2 ≤M U X p . From Young's inequality, see [1] page 92, we can choose a constant δ > 0 such that with C δ > 0. Then we take δ > 0 such that Θ > δM p C, and define the following constant Remark 3. Note that Then, consider the following additional assumption: e σs g 1 (s) p p + g 2 (s) p p ,Γ ds < +∞, ∀ t ∈ R.
Lemma 5.1 ensures that the multivalued process {U (t, τ )} is pullback dissipative in C λ (X 2 ), with the following pullback absorbing family.
Lemma 5.2. Suppose hypotheses of Lemma 5.1 and for each t ∈ R define where Then D = {D(t)} t∈R is a pullback absorbing family for the multivalued process Note that when the constant σ in the estimate (16) is positive we can ensure the stability of the solution for τ ∈ R small enough. This kind of property is proved in other works asking for some restrictions on constants like K i 's in (F3), see for instance [2,3,15,16,17] and [19]. Here, according to the handling presented at the beginning of the section, we obtain an appropriate σ for each choice of K i 's. It means that the existence of a pullback absorbing family is independent of the choice of K i 's. Lemma 5.3. Let λ > Θ δM p , then the multivalued process {U (t, τ )} is pullback asymptotically upper-semicompact in C λ (X 2 ).
Proof. Let t 0 ∈ R fixed, and let {U n (t 0 , τ n , Ψ n )} n∈N be a sequence of weak solutions of Problem (P ) with {Ψ n } n∈N ⊂ C λ (X 2 ) a sequence of initial conditions in {τ n } n∈N ⊂ R, respectively, and τ n → −∞. Without loss of generality, we may assume that τ n < t 0 for all n ∈ N.
Consider {U t0 n } and we will show that such sequence is precompact in C λ (X 2 ) in two steps. See [15,17] for examples of the same technique.
Step 1. We will show that there exist a function W : (−∞, 0] → X 2 and a subsequence of {U t0 n }, relabelled the same, such that U t0 n → W in C([−T, 0]; X 2 ) for every T > 0.
with Y 0 n = U t0−T n and Y T n = U t0 n . From (17) we have that Y 0 n 2 C λ (X 2 ) ≤ R(t 0 , T ), ∀ n ≥ n 0 (t 0 , T ), and, from a priori estimate (3) it is possible to find K(t 0 , T ) such that Y n L p (0,T ;V p ) ≤ K(t 0 , T ). (20) Thanks to these estimates there exists Y ∈ L ∞ (0, T ; X 2 ) ∩ L 2 (0, T ; V p ) such that Note that, from hypothesis (F3) there existsK =K(K 1 , K 2 ) > 0 such that and (19) ensures thatF(t, Y t n ) is bounded in L p (0, T ; (V p ) * ), and from (20) we have that operator β p Y n is bounded in the same space. Then, as it was done in the proof of Theorem 4.3, there exists ∂ t Y ∈ L p (0, T ; (V p ) * ) such that From (21), (23) and Theorems 1.4 and 1.5 in page 32 of [4], we have Y n → Y in L p (τ, T ; X 2 ), Y n → Y in C([τ, T ]; X 2 ).