Pullback attractors for a class of non-autonomous thermoelastic plate systems

In this article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ \begin{cases} u_{tt} +\eta\Delta^2 u+a(t)\Delta\theta=f(t,u),&t>\tau,\ x\in\Omega,\\ \theta_t-\kappa\Delta \theta-a(t)\Delta u_t=0,&t>\tau,\ x\in\Omega, \end{cases} $$ subject to boundary conditions $$ u=\Delta u=\theta=0,\ t>\tau,\ x\in\partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partial\Omega$ is assumed to be a $\mathcal{C}^4$-hypersurface, $\eta>0$ and $\kappa>0$ are constants, $a$ is an H\"older continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.


Introduction
This paper is concerned with the qualitative behavior of non-autonomous dynamical systems generated by a non-autonomous thermoelastic plate system, in particular as described by their pullback attractors. The problem that we consider in this paper is the following.
Here, we assume that there exist positive constants a 0 and a 1 such that (1.3) 0 < a 0 a(t) a 1 , ∀t ∈ R.
Below we give conditions under which the non-autonomous problem (1.1)-(1.2) is locally and globally well posed in some space that we will specify later. To that end we must assume some growth condition on the nonlinearity f .
To obtain the global existence of solutions and the existence of pullback attractor we assume that f : R 2 → R is locally Lipschitz in the second variable, and it is a dissipative nonlinearity in the second variable (1.5) lim sup |s|→∞ f (t, s) s < λ 1 , uniformly in t ∈ R, where λ 1 > 0 is the first eigenvalue of negative Laplacian operator with zero Dirichlet boundary condition. Due to Sobolev embedding we need to assume that the function f satisfies subcritical growth condition; that is, (1.6) |f s (t, s)| C(1 + |s| ρ−1 ), ∀s ∈ R, where 1 ρ < N N −4 , with N 5, and C > 0 independent of t ∈ R. We will justify these restrictions later in the paper. If N = 2, 3, 4, we suppose the growth condition (1.6) with ρ 1.
Although we get global solutions for initial-boundary value problem (1.1)-(1.2) in the space H 2 (Ω) × L 2 (Ω) × L 2 (Ω) with the nonlinearity f satisfying the growth condition (1.6) with 1 ρ < N +4 N −4 for N 5, see Section 4, to study the asymptotic behavior of solutions we only consider the condition (1.6) with 1 ρ < N N −4 for N 5, see Section 5. The purpose of this paper is to prove, under suitable assumptions, local and global wellposedness (using the uniform sectorial operators theory) of the non-autonomous problem (1.1)-(1.2), the existence of pullback attractors and uniform bounds for these pullback attractors.
It is very well known that the model in (1.1) describes the small vibrations of a homogeneous, elastic and thermal isotropic Kirchhoff plate. In the literature the initial boundaryvalue problem (1.1)-(1.2) has been extensively discussed for several authors in differents contexts. For instance, Baroun et al. in [5] studied the existence of almost periodic solutions for an evolution system like (1.1), Liu and Renardy [17] proved that the linear semigroup defined by system (1.1) with f ≡ 0 with clamped boundary condition for u and Dirichlet boundary condition for θ is analytic. The typical difficulties in thermoelasticity comes from the boundary condition, which make more complicated the task of getting estimates to show the exponential stability of the solutions or analyticity of the corresponding semigroup. In that direction we have the works of Liu and Zheng [18], Lasiecka and Triggiani [16] to free -clamped boundary condition. In this last work the authors show the exponential stability and analyticity of the semigroup associated with the system (1.1). We refer to the book of Liu and Zheng [19] for a general survey on those topics.
To formulate the non-autonomous problem (1.1)-(1.2) in the nonlinear evolution process setting, we introduce some notations. Here, we denote X = L 2 (Ω) and Λ : D(Λ) ⊂ X → X the biharmonic operator defined by D(Λ) = {u ∈ H 4 (Ω); u = ∆u = 0 on ∂Ω} and Λu = (−∆) 2 u, ∀u ∈ D(Λ), then Λ is a positive self-adjoint operator in X with compact resolvent and therefore −Λ generates a compact analytic semigroup on X (that is, Λ is a sectorial operator, in the sense of Henry [15]). Denote by X α , α > 0, the fractional power spaces associated with the operator Λ; that is, X α = D(Λ α ) endowed with the graph norm. With this notation, we have X −α = (X α ) ′ for all α > 0, see Amann [1] for the characterization of the negative scale. It is of special interest the case α = 1 2 , since −Λ 1 2 is the Laplacian operator with homogeneous Dirichlet boundary conditions.
If we denote v = u t , then we can rewrite the non-autonomous problem (1.1)-(1.2) in the matrix form where w = w(t) for all t ∈ R, and w 0 = w(τ ) are given by and, for each t ∈ R, the unbounded linear operator A (a) (t) : We define the nonlinearity F by where f e (t, u) is the Nemitskiȋ operator associated with f (t, u), t ∈ R, that is, The map f e (t, u) is Lipschitz continuous in bounded subsets of X 1 2 uniformly in t ∈ R. This paper is organized as: In Section 2 we recall concepts and results about problems singularly non-autonomous, including results on existence of pullback attractors. In Section 3 we deal with the linear problem associated (1.1)-(1.2). Section 4 is devoted to study the existence of local and global solutions in some appropriate space. Finally, in Section 5 we present results on dissipativeness of thermoelastic equation and existence of pullback attractors for (1.1)-(1.2).

Singularly non-autonomous abstract problem
Throughout the paper, L(Z) will denote the space of linear and bounded operators defined in a Banach space Z. Let A(t), t ∈ R, be a family of unbounded closed linear operators defined on a fixed dense subspace D of Z.

2.1.
Singularly non-autonomous abstract linear problem. Consider the singularly non-autonomous abstract linear parabolic problem of the form We assume that (a) The operator A(t) : D ⊂ Z → Z is a closed densely defined operator (the domain D is fixed) and there is a constant C > 0 (independent of t ∈ R) such that ; for all λ ∈ C with Reλ 0.
To express this fact we will say that the family A(t) is uniformly sectorial. (b) There are constants C > 0 and ǫ 0 > 0 such that, for any t, τ, s ∈ R, To express this fact we will say that the map R ∋ t → A(t) is uniformly Hölder continuous.
Denote by A 0 the operator A(t 0 ) for some t 0 ∈ R fixed. If Z α denotes the domain of A α 0 , α > 0, with the graph norm and Z 0 := Z, denote by {Z α ; α 0} the fractional power scale associated with A 0 (see Henry [15]).
Next we recall the definition of a linear evolution process associated with a family of operators {A(t) : t ∈ R}. 2) L(t, σ)L(σ, τ ) = L(t, τ ), for any t σ τ, is called a linear evolution process ( process for short) or family of evolution operators.
If the operator A(t) is uniformly sectorial and uniformly Hölder continuous, then there exists a linear evolution process {L(t, τ ) : t τ ∈ R} associated with A(t), which is given by The evolution process {U(t, τ ) : t τ ∈ R} satisfies the following condition: where 0 β α < 1 + ǫ 0 . For more details see [12] and [20].

2.2.
Existence of pullback attractors. In this subsection we will present basic definitions and results of the theory of pullback attractors for nonlinear evolution process. For more details we refer to [7], [8], [11] and [13]. We consider the singularly non-autonomous abstract parabolic problem where the operator A(t) is uniformly sectorial and uniformly Hölder continuous and the nonlinearity g satisfies conditions which will be specified later. The nonlinear evolution process {S(t, τ ) : t τ ∈ R} associated with A(t) is given by be a continuous function. We say that a continuous function u : We can now state the following result, proved in [12, Theorem 3.1].

Theorem 2.3. Suppose that the family of operators A(t) is uniformly sectorial and uniformly Hölder continuous in
, is a Lipschitz continuous map in bounded subsets of X α , then, given r > 0, there is a time t 0 > 0 such that for all u 0 ∈ B X α (0; r) there exists a unique solution of the problem (2.2) starting in u 0 and defined in [τ, τ + t 0 ]. Moreover, such solutions are continuous with respect the initial data in B X α (0; r).
We start remembering the definition of Hausdorff semi-distance between two subsets A and B of a metric space (X, d): Next we present several definitions about theory of pullback attractors, which can be founded in [7,11,13].
τ ∈ R} be an evolution process in a metric space X.
Definition 2.5. The pullback orbit of a subset B ⊂ X relatively to the evolution process Definition 2.8. We say that a family of bounded subsets {B(t) : t ∈ R} of X is pullback absorbing for the evolution process {S(t, τ ) : t τ ∈ R}, if for each t ∈ R and for any bounded subset B of X, there exists τ 0 (t, B) t such that Definition 2.9. A family of subsets {A(t) : t ∈ R} of X is called a pullback attractor for the evolution process is compact for all t ∈ R, and pullback attracts bounded subsets of X at time t, for each t ∈ R.
In applications, to prove that a process has a pullback attractor we use the Theorem 2.11, proved in [7], which gives a sufficient condition for existence of a compact pullback attractor. For this, we will need the concept of pullback strongly bounded dissipativeness. Definition 2.10. An evolution process {S(t, τ ) : t τ ∈ R} in X is pullback strongly bounded dissipative if, for each t ∈ R, there is a bounded subset B(t) of X which pullback absorbs bounded subsets of X at time s for each s t; that is, given a bounded subset B of X and s t, there exists τ 0 (s, B) such that S(s, τ )B ⊂ B(t), for all τ τ 0 (s, B). Now we can present the result which guarantees the existence of pullback attractors for non-autonomous problems, see [7].
Theorem 2.11. If an evolution process {S(t, τ ) : t τ ∈ R} in the metric space X is pullback strongly bounded dissipative and pullback asymptotically compact, then {S(t, τ ) : The next result gives sufficient conditions for pullback asymptotic compactness, and its proof can be found in [7]. Theorem 2.12. Let {S(t, s) : t s ∈ R} be a pullback strongly bounded evolution process such that S(t, s) = L(t, s)+U(t, s), where there exist a non-increasing function k : R + ×R + → R, with k(σ, r) → 0 when σ → ∞, and for all s t and x ∈ X with x r, L(t, s)x k(t − s, r), and U(t, s) is compact. Then, the family of evolution process {S(t, s) : t s ∈ R} is pullback asymptotically compact.

Linear Analysis
In this section we consider the linear problem associated with (1.1)-(1.2), in this case we consider the singularly non-autonomous linear parabolic problem where w, w 0 are defined in (1.8) and the linear unbounded operador A (a) is defined by (1.9)-(1.10).
It is not difficult to see that det(A (a) (t)) = ηκΛ 3 2 and 0 ∈ ρ(A (a) (t)) for any t ∈ R. Moreover we have that the operator The following equality holds Proof: Recall first that H is the completion of the normed space (H, A −1 a (t) · H ). Since there are positive constants C 1 and C 2 such that for any (u . Below is the proof of this last statement. Let [u v θ] T ∈ L 2 (Ω) × H −2 (Ω) × H −2 (Ω), and note that .
Note that the operator A a (t) can be extended to its closed H−realization (see Amann [1] p. 262), which we will still denote by the same symbol so that A a (t) considered in H is then sectorial positive operator (see [5]). Our next concern will be to obtain embedding of the spaces from the fractional powers scale H α , α 0, generated by (A a (t), H). (or simply L (a) (t, τ )) associated with the operator A (a) (t), that is given by The linear evolution operator {L (a) (t, τ ) : t τ ∈ R} satisfies the condition (2.1).

Existence of global solutions
In this section we study the existence of global solutions for (1.7). It is not difficult to prove the following result, see for instance Lemma 2.4 in [9].
The next result also can be found in [9], see Lemma 2.5, we present the proof for the sake of completeness.
Remark 4.2. If H =: Y 0 , then the fractional power spaces Y α , α ∈ [0, 1], are given by where [·, ·] α denotes the complex interpolation functor (see [21]).  Since τ can be chosen uniformly in bounded subsets of Y , the solutions which do not blow up in Y must exist globally. Alternatively, we obtain a uniform in time estimate of (u(t), ∂ u (t), θ(t)) Y , such estimate is needed to justify global solvability of the problem (1.7) in Y = H 2 (Ω) × L 2 (Ω) × L 2 (Ω). Consider the original system (1.1) (or (1.7) in Y ). Multiplying the first equation in (1.1) by u t , and the second equation in (1.1) by θ, we get the system and integrating over Ω we obtain for any t > τ . Note that Combining (4.2) with (4.3) we have d dt for any t > τ . The total energy of the system E(t) associated with the solution (u(t), ∂ t u(t), θ(t)) of (1.1)-(1.2) is defined by This identity says that the function t → E(t) becomes monotone decreasing.
We obtain (from (1.5)) that for each ε > 0, there exists C ε > 0 such that then the property E(t) E(τ ) offers an a priori estimate of the solution (u(t), ∂ t u(t), θ(t)) in H 2 (Ω) × L 2 (Ω) × L 2 (Ω). In fact, and, if we choose 0 < ε 0 < 1 2c , we get a boundedness as desired, that is, With this, we ensure that there exists a global solution w(t) for Cauchy problem (1.7) in Y and it defines an evolution process {S (a) (t, τ ) : t τ ∈ R}, that is, According to [12] (4.7) where {L (a) (t, τ ) : t τ ∈ R} is the linear evolution process associated with the homogeneous problem (1.7).

Dissipativeness of the thermoelastic equation
In this section we combine the arguments from [2], [3], [4], [6] and [14] in order to prove the existence of pullback attractors for (1.1)-(1.2). To achieve our purpose we consider the functionals From this we define an energy functional where (5.4) 0 < δ 1 < δ 2 < 1 and M > 0 will be fixed later. We recall that E(t) is decreasing since E ′ (t) 0 from (4.4).
Proof: In the following, C 0 and C 1 will denote positive constants depending on the embedding constants and initial data, respectively, as far it is necessary. Note that Due to (5.3) and Poincaré's inequality we have where λ 1 is the first eigenvalue of negative Laplacian operator with zero Dirichlet boundary condition. Furthermore, and from (1.3) we get and by Young's inequality To deal with the integral term, just notice that from dissipativeness condition (1.5), there exists C ν > 0 such that Ω f (t, u)udx ν u 2 L 2 (Ω) + C ν , and thus, where µ 1 > 0 is the embedding constant for ∇u 2 , and ν is chosen such that 0 < ν < λ 1 η µ 1 .
This concludes the proof of the theorem. Again, by (5.13) together with Remark 5.1, we conclude where γ 1 = γ 1 (L(τ )) > 0 and γ 2 > 0. Proof: From estimate (5.16) it is easy to check that the evolution process {S (a) (t, τ ) : t τ ∈ R} associated with (1.1)-(1.2) is pullback strongly bounded. Hence, applying the same ideas of the proofs of the Theorem 5.1 and Theorem 5.2, we obtain that the family of evolution process {S (a) (t, τ ) : t τ ∈ R} is pullback asymptotically compact (see Theorem 2.12). In fact, from (4.7) we write With the same arguments used in the proof of the Theorem 5.1 with f ≡ 0 in (1.1) and with the functionals and L(t) = ME(t) + δ 1 φ(t) + δ 2 ψ(t) where φ is defined in (5.1) and ψ is defined in (5.2), we get from (5.5) that there exist c 1 > 0 such that L ′ (t) −c 1 E(t) and from arguments used in the proof of the Theorem 5.2 with f ≡ 0 in (1.1), by (5.13) we get c 2 , c 3 > 0 such that c 2 E(t) L(t) c 3 E(t) and hence E ′ (t) −c 0 E(t) for some c 0 > 0. This ensures that exist constants K, α > 0 such that L (a) (t, τ ) L(Y ) Ke −α(t−τ ) , for all t τ.