Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids

Abstract

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

Keywords

Mixture of solids time delay quasi-stability global attractors upper-semicontinuity

1. Introduction

The theory of mixture of solids was given by Bowen [6], achieving a connection between the microscopic deformation approach to the macroscopic flow response in porous media. Several works have been dedicated to the study of important effects in the theory of mixture of solids, such as the relative effect of the fluid fractions that fills the porous medium, known as saturation, studied in [3, 14–16]. So far, there are many authors having studied the asymptotic behavior of solutions for mixture binary of solids incorporating dissipative effects. For exemple, Alves and Rivera [1] considered the following one-dimensional system composed of a mixture of two thermoelastic solids $\begin{matrix} (1.1) & \{\begin{matrix} ρ_{1} u_{t t} - a_{11} u_{x x} - a_{12} w_{x x} + α (u - w) - β_{1} θ_{x} = 0 & in (0, L) \times (0, \infty), \\ ρ_{2} w_{t t} - a_{12} u_{x x} - a_{22} w_{x x} - α (u - w) - β_{2} θ_{x} = 0 & in (0, L) \times (0, \infty), \\ c θ_{t} - κ θ_{x x} - β_{1} u_{x t} - β_{2} w_{x t} = 0 & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ where u, w and θ represent the displacement of typical particles in time t, and the temperature. The constants $ρ_{1}$ , $ρ_{2}$ , κ, c and α are positive, and in addition, the coefficients satisfy $a_{11} > 0$ and $a_{11} a_{22} - a_{12}^{2} > 0$ . The core result is the exponential decay of the solution semigroup. Precisely, the exponential decay holds if and only if, the relation $β_{2} (β_{1} ρ_{2} a_{11} + β_{2} ρ_{1} a_{12}) \neq β_{1} (β_{2} ρ_{1} a_{22} + β_{1} ρ_{2} a_{12})$ is satisfied. Recently, Rivera, Naso and Quintanilla [24] studied the asymptotic behavior of the solution for mixtures given by $\begin{matrix} (1.2) & \{\begin{matrix} ρ_{1} u_{t t} - a_{11} u_{x x} - a_{12} w_{x x} + α (u - w) + β_{1} θ_{1, x} + β_{2} θ_{1, x} = 0 & in (0, L) \times (0, T), \\ ρ_{2} w_{t t} - a_{12} u_{x x} - a_{22} w_{x x} - α (u - w) + γ_{1} θ_{1, x} + γ_{2} θ_{1, x} = 0 & in (0, L) \times (0, T), \\ b_{1} θ_{1, t} + b_{2} θ_{2, t} - K_{11} θ_{1, x x} - K_{12} θ_{2, x x} + β_{1} u_{x t} + β_{2} w_{x t} + a (θ_{1} - θ_{2}) = 0 & in (0, L) \times (0, T), \\ b_{2} θ_{1, t} + b_{3} θ_{2, t} - K_{21} θ_{1, x x} - K_{22} θ_{2, x x} + γ_{1} u_{x t} + γ_{2} w_{x t} - a (θ_{1} - θ_{2}) = 0 & in (0, L) \times (0, T), \end{matrix} \end{matrix}$ where the components had two different temperatures. Exponential decay was proven for a generic class of materials. However, for restricted conditions and considering a Neumann boundary condition, the authors concluded that the system is not exponentially stable in general; they proved that the solution decays polynomially at rate $t^{- 1 / 2}$ . The authors also demonstrated that this decay is optimal for the model studied. Using a Dirichlet boundary condition, the decay rate was $t^{- 1 / 6}$ .

On the other hand, Quintanilla [28] analyzed the exponential stability in the linear isothermal theory of swelling porous elastic soils. The author considered the problem given by $\begin{matrix} (1.3) & \{\begin{matrix} ρ_{z} z_{t t} - a_{1} z_{x x} - a_{2} u_{x x} + ξ (z_{t} - u_{t}) - μ_{z} z_{x x t} = 0 & in (0, L) \times (0, \infty), \\ ρ_{u} u_{t t} - μ u_{x x} - a_{2} z_{x x} - ξ (z_{t} - u_{t}) = 0 & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ where the dependent variables $z = z (x, t)$ and $u = u (x, t)$ represent the displacement of the fluid and the elastic solid material respectively. He proved that the dissipative terms $\pm ξ (z_{t} - u_{t})$ and $μ_{z} z_{x x t}$ are sufficient to establish exponential stability using the energy method.

With respect to the stability of the nonlinear system, Santos et al. [34] considered the following binary mixture problem of solids with nonlinear damping and source terms $\begin{matrix} (1.4) & \{\begin{matrix} ρ_{1} u_{t t} - a_{11} u_{x x} - a_{12} w_{x x} + g_{1} (u_{t}) = f_{1} (u, w), & in (0, L) \times (0, T), \\ ρ_{2} w_{t t} - a_{12} u_{x x} - a_{22} w_{x x} + g_{2} (w_{t}) = f_{2} (u, w), & in (0, L) \times (0, T), \end{matrix} \end{matrix}$ with Dirichlet boundary condition on both u and w. The authors show the existence of local and global weak solutions, uniqueness of weak solutions, and continuous dependence of initial data. Under some restrictions on the parameters, they also proved that every weak solution to system blows-up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via potential well theory. Specifically, they proved the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.

The study of the delay effect in the stabilization of hyperbolic systems has been made in recent years and many researchers have shown the so called destabilizing effect. In many cases it was shown that time delays can be a source of instability, same with an arbitrary small delay. That is to say, the time delay may destabilize an evolution equation which is uniformly asymptotically stable in the absence of delay unless some control terms have been used. The first contributions in that direction appear with the papers due to Datko et al. [10–13]. In particular, Datko [10] considered the equation given by $\begin{matrix} (1.5) & u_{t t} - u_{x x} + 2 u_{t} (x, t - τ) = 0, (x, t) \in (0, 1) \times (0, + \infty) . \end{matrix}$ He showed that the time delay (represented by $t - τ$ for $τ > 0$ ) in the damping given by velocity term can destabilize the system. The same result was obtained by Datko et al. [13] for time delay acting on boundary control and the authors showed that the well-behaved hyperbolic system turns in a chaotic system from which they concluded that delay becomes a source of instability. In particular, Datko [11] presented examples of hyperbolic equations which change for an unstable regime by small time delays in the boundary feedback control. On the other hand, in order to stabilize a hyperbolic system with time delay terms, it is necessary to add control terms. In that direction, Nicaise and Pignotti [25] (see also the same authors in [26]) considered the following system $\begin{array}{l} (1.6) & u_{t t} - Δ u = 0, x \in Ω, t > 0, \\ (1.7) & u (x, t) = 0, x \in Γ_{0}, t > 0, \\ (1.8) & \frac{\partial u}{\partial ν} (x, t) = μ_{1} u_{t} (x, t) + μ_{2} u_{t} (x, t - τ), x \in Γ_{1}, t > 0, \end{array}$ where u denotes wave propagations in a bounded domain region $Ω \subset R^{n}$ with a smooth boundary Γ which is divided in two closed and disjoint parts $Γ_{0}$ and $Γ_{1}$ , i.e., $Γ = Γ_{0} \cup Γ_{1}$ and $Γ_{0} \cap Γ_{1} = \emptyset$ . Moreover, ν denotes the outer unit normal vector, $\frac{\partial u}{\partial ν}$ is the normal derivative, $μ_{1}$ and $μ_{2}$ are two real numbers and $(u_{0}, u_{1}, g_{0})$ denotes the initial data. The authors prove that under the assumption $\begin{matrix} (1.9) & μ_{2} < μ_{1}, \end{matrix}$ the total energy of solutions is exponentially stable. However, if $μ_{2} > μ_{1}$ , the system (1.6)–(1.8) is unstable.

In the recent works of Ramos et al. [29, 30] who considers the swelling system of porous elastic soils given by $\begin{matrix} (1.10) & \{\begin{matrix} ρ_{z} z_{t t} - a_{1} z_{x x} - a_{2} u_{x x} + ξ_{1} z_{t} + ξ_{2} z_{t} (x, t - τ) = 0 & in (0, L) \times (0, \infty), \\ ρ_{u} u_{t t} - a_{3} u_{x x} - a_{2} z_{x x} + γ (t) g (u_{t}) = 0 & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ and presented several results related to the stability of the system. For example, in [29], it is proven that the system (1.10) with $ξ_{1} = ξ_{2} = 0$ decays exponentially, using a multiplier method and some properties of convex functions. In [30] it is proven that the system (1.10) with $γ (t) = 0$ for all $t ⩾ 0$ , is exponentially stable since $| ξ_{2} | < ξ_{1}$ .

On the other hand, concerning global attractors for binary mixture of solids with nonlinear damping and source terms, we mention the work of Santos and Freitas [33] who proved the existence of compact global attractor with finite fractal dimension and the existence of generalized exponential attractor. The existence and upper-semicontinuity of global attractors for binary mixtures of solids with fractional damping been considered by Freitas et al. in [18]. Finally, Freitas et al. [17] studied the pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. One point that we verified that has not yet been addressed in the literature is the mixture of solids, considering the small delays effects, as well as the existence of a global and exponential attractors. Thus, the present article aims to be a new contribution to this subject.

In this paper, we study the binary mixture problem with delay term given by $\begin{matrix} (1.11) & \{\begin{matrix} ρ_{z} z_{t t} - a_{1} z_{x x} - a_{2} u_{x x} + z_{t} + f_{1} (z, u) = h_{1}, & in (0, L) \times (0, T), \\ ρ_{u} u_{t t} - a_{3} u_{x x} - a_{2} z_{x x} + u_{t} + ϵ u_{t} (x, t - τ) + f_{2} (z, u) = h_{2}, & in (0, L) \times (0, T), \end{matrix} \end{matrix}$ with boundary and initial conditions $\begin{matrix} (1.12) & \{\begin{matrix} z (0, t) = z_{x} (L, t) = u (0, t) = u_{x} (L, t) = 0, & t > 0, \\ z (x, 0) = z_{0} (x), z_{t} (x, 0) = z_{1} (x), & x \in (0, L), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in (0, L), \end{matrix} \end{matrix}$ where z and u represent the displacement of each constituent, $ρ_{z} > 0$ and $ρ_{u} > 0$ are the mass densities, ϵ is a positive constant small enough, $f_{1}$ and $f_{2}$ are nonlinear source terms and the functions $h_{1}$ and $h_{2}$ are external forces. Here $a_{i}$ , $i = 1, 2, 3$ , satisfy the relation $\begin{matrix} a_{2}^{2} < a_{1} a_{3} . \end{matrix}$

The main contributions in this paper are threefold:

We study the asymptotic analysis in the sense of compact global attractors of a binary mixture of solids with small delay effects for the first time in the literature.

Instead of showing the existence of an absorbing set, we prove the system is gradient and asymptotic smoothness, and hence obtain the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions.

We use the recent quasi-stability theory developed in [7, 9] directly on a bounded positively invariant set. This new and powerful theory implies at once the smoothness and finite fractal dimension of the attractor as well as the existence of exponential attractors.

The upper semicontinuity of global attractors with time delay perturbation is a new contribution. In this paper, we prove the upper semicontinuity of global attractors as delay term disappear, that is, we prove that the family of global attractors $A_{ϵ}$ associated to the problem (1.11)–(1.12) converges to the global attractor $A_{0}$ associated to the problem (1.11)–(1.12) with $ϵ = 0$ .

This paper is organized as the following. In Section 2, the existence and uniqueness of weak and strong solutions are established by using the theory of monotone operators (see [4, 8]). In Section 3, key definitions and known results concerning the dynamical systems and global attractors are introduced. In Section 4, we summarize the main results. Section 5 is devoted to prove the existence of attractors and their properties. In the first subsection we prove the stabilizability inequality and quasi-stability of the system. The second subsection is devoted to prove the that the system is gradient. In the third subsection, we prove the Theorem 4.1. More precisely, the existence of finite fractal global attractors with smoothness properties and the existence of a generalized fractal exponential attractor. Section 6 is dedicated to prove the upper semicontinuity of global attractors.

2. Preliminaries and well-posedness

In this section, the existence and uniqueness of weak and strong solutions of the system (1.11)–(1.12) is studied. Beforehand, we present preliminaries including notations, assumptions and technical lemmas.

2.1. Preliminaries

In order to obtain the well-posedness of the problem (1.11)–(1.12), we introduce the following variable by [27, 32], $\begin{matrix} (2.1) & θ (x, y, t) = u_{t} (x, t - τ y) in (0, L) \times (0, 1) \times (0, T), \end{matrix}$ that satisfies the differential identity $\begin{matrix} (2.2) & τ θ_{t} (x, y, t) + θ_{y} (x, y, t) = 0 in (0, L) \times (0, 1) \times (0, T) . \end{matrix}$ Therefore, the system (1.11)–(1.12) takes the following form $\begin{matrix} (2.3) & \{\begin{matrix} ρ_{z} z_{t t} - a_{1} z_{x x} - a_{2} u_{x x} + z_{t} + f_{1} (z, u) = h_{1}, & in (0, L), \times (0, T), \\ ρ_{u} u_{t t} - a_{3} u_{x x} - a_{2} z_{x x} + u_{t} + ϵ θ (x, 1, t) + f_{2} (z, u) = h_{2}, & in (0, L) \times (0, T), \\ τ θ_{t} + θ_{y} = 0, & in (0, L) \times (0, 1) \times (0, T), \end{matrix} \end{matrix}$ with boundary and initial conditions $\begin{matrix} (2.4) & \{\begin{matrix} z (0, t) = z_{x} (L, t) = u (0, t) = u_{x} (L, t) = 0, & t > 0, \\ z (x, 0) = z_{0} (x), z_{t} (x, 0) = z_{1} (x), & x \in (0, L), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in (0, L), \\ θ (x, y, 0) = f_{0} (x, - τ y), & (x, y) \in (0, L) \times (0, 1) . \end{matrix} \end{matrix}$

We use throughout this paper the standard Lebesgue spaces $L^{p} (0, L)$ , $p ⩾ 1$ , with the norm denoted by $‖ \cdot ‖_{p}$ . We denote by $⟨ \cdot, \cdot ⟩$ the inner product in $L^{2} (0, L)$ . Let us consider the Sobolev space $\begin{matrix} H_{*}^{1} (0, L) = {u \in H^{1} (0, L) : u (0) = 0} . \end{matrix}$ Since $u (0) = 0$ , the Poincaré’s inequality holds $\begin{matrix} (2.5) & λ_{0} ‖ u ‖_{2}^{2} ⩽ ‖ u_{x} ‖_{2}^{2}, \forall u \in H_{*}^{1} (0, L), \end{matrix}$ where $λ_{0} > 0$ is the Poincaré’s constant. Therefore, $‖ u ‖_{H_{*}^{1} (0, L)} : = ‖ u_{x} ‖_{2}$ is an equivalent norm in $H_{*}^{1} (0, L)$ .

We consider the following list of assumptions.

Assumption 2.1.
We assume that
The external forces $h_{1}, h_{2} \in L^{2} (0, L)$ .

There exists a function $F \in C^{2} (R^{2})$ such that $\begin{matrix} (2.6) & \nabla F = (f_{1}, f_{2}) . \end{matrix}$

There exist $q ⩾ 1$ and $C > 0$ such that $\begin{matrix} (2.7) & | \nabla f_{j} (z, u) | ⩽ C (| z |^{q - 1} + | u |^{q - 1} + 1), j = 1, 2 . \end{matrix}$

There exist constants $η ⩾ 0$ , $m_{F} > 0$ such that $\begin{matrix} (2.8) & F (z, u) ⩾ - η (| z |^{2} + | u |^{2}) - m_{F} . \end{matrix}$ with $0 ⩽ η < \frac{1}{2 η_{0}}$ , where $η_{0} > 0$ is a constant specified later in (2.13). Moreover, $\begin{matrix} (2.9) & \nabla F (z, u) \cdot (z, u) - F (z, u) ⩾ - η (| z |^{2} + | u |^{2}) - m_{F} . \end{matrix}$ In addition, since $ϵ \in [0, 1)$ we have $\begin{matrix} (2.10) & τ ϵ ⩽ τ ⩽ τ (2 - ϵ) . \end{matrix}$

Define the phase space $\begin{matrix} (2.11) & H = H_{}^{1} (0, L) \times H_{}^{1} (0, L) \times L^{2} (0, L) \times L^{2} (0, L) \times L^{2} ((0, L) \times (0, 1)) . \end{matrix}$ with the following inner product $\begin{matrix} {(U, \tilde{U})}_{H} & = ρ_{z} ⟨ ϕ, \tilde{ϕ} ⟩ + ρ_{u} ⟨ φ, \tilde{φ} ⟩ x + χ ⟨ u_{x}, {\tilde{u}}_{x} ⟩ + ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x}, \frac{a_{2}}{\sqrt{a_{1}}} {\tilde{u}}_{x} + \sqrt{a_{1}} {\tilde{z}}_{x} ⟩ \\ + τ \int_{0}^{L} \int_{0}^{1} θ (x, y) \tilde{θ} (x, y) d x d y, \end{matrix}$ where $U = (z, u, ϕ, φ, θ)$ , $\tilde{U} = (\tilde{z}, \tilde{u}, \tilde{ϕ}, \tilde{φ}, \tilde{θ}) \in H$ and $\begin{matrix} χ = a_{3} - \frac{a_{2}^{2}}{a_{1}} . \end{matrix}$ The norm induced by the inner product is then $\begin{matrix} ‖ U ‖_{H}^{2} & = ρ_{z} ‖ ϕ ‖^{2} + ρ_{u} ‖ φ ‖_{2} + χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2} \\ + τ \int_{0}^{L} \int_{0}^{1} {| θ (x, y) |}^{2} d x d y . \end{matrix}$ We observe that there exists a constant $κ_{0} > 0$ such that $\begin{matrix} (2.12) & ‖ z_{x} ‖_{2}^{2} + ‖ u_{x} ‖_{2}^{2} ⩽ κ_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) . \end{matrix}$ Combining (2.5) and (2.12), there exists a constant $η_{0} > 0$ such that $\begin{matrix} (2.13) & ‖ z ‖_{2}^{2} + ‖ u ‖_{2}^{2} ⩽ η_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) . \end{matrix}$
2.2. Well-posedness

Next, we establishes the well-posedness for the problem (2.3)–(2.4). Tho this end, we rewritten the system (2.3)–(2.4) in the following Cauchy problem $\begin{matrix} (2.14) & \{\begin{matrix} \frac{d U (t)}{d t} + A U (t) = F (U (t)), & t > 0, \\ U (0) = U_{0} \in H, \end{matrix} \end{matrix}$ where $U_{0} = (z_{0}, u_{0}, z_{1}, u_{1}, θ_{0}) \in H$ and the linear operator $A : D (A) \subset H \to H$ is given by $\begin{matrix} (2.15) & A U = (\begin{matrix} - ϕ \\ - φ \\ - \frac{1}{ρ_{z}} {(a_{1} z + a_{2} u)}_{x x} + \frac{1}{ρ_{z}} ϕ \\ - \frac{1}{ρ_{u}} {(a_{3} u + a_{2} z)}_{x x} + \frac{1}{ρ_{u}} φ + \frac{ϵ}{ρ_{u}} θ (1) \\ \frac{1}{τ} θ_{y} \end{matrix}) . \end{matrix}$ Here the domain of $A$ is $\begin{matrix} D (A) : = & {U = (z, u, ϕ, φ, θ) \in H : z, u \in H^{2} (0, L), ϕ, φ \in H_{*}^{1} (0, L), \\ z_{x} (L) = u_{x} (L) = 0, θ \in L^{2} (0, L; H_{*}^{1} (0, 1)), φ = θ (\cdot, 0)} . \end{matrix}$ The forcing terms are represented by a nonlinear function $F : H \to H$ defined by $\begin{matrix} F (U) : = (\begin{matrix} 0 \\ 0 \\ \frac{1}{ρ_{z}} (h_{1} - f_{1} (z, u)) \\ \frac{1}{ρ_{u}} (h_{2} - f_{2} (z, u)) \\ 0 \end{matrix}) . \end{matrix}$

Now, we introduce the precise definition of weak and strong solutions for the problem (2.3)–(2.4).

Definition 2.2.
A function $U = (z, u, z_{t}, u_{t}, θ) \in C ([0, \infty); H)$ , is a weak solution to (2.3)–(2.4) with initial condition $\begin{matrix} U (0) = (z_{0}, u_{0}, z_{1}, u_{1}, f_{0}) = U_{0} \end{matrix}$ if for all test functions $φ, ψ \in H_{}^{1} (0, L)$ and almost every $t > 0$ $\begin{matrix} (2.16) & \begin{matrix} ρ_{z} \frac{d}{d t} ⟨ z_{t}, φ ⟩ + ρ_{u} \frac{d}{d t} ⟨ u_{t}, ψ ⟩ + χ ⟨ u_{x}, ψ_{x} ⟩ + ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x}, \frac{a_{2}}{\sqrt{a_{1}}} ψ_{x} + \sqrt{a_{1}} φ_{x} ⟩ \\ + ⟨ z_{t}, φ ⟩ + ⟨ u_{t}, ψ ⟩ + ϵ ⟨ θ (1, t), ψ ⟩ + ⟨ f_{1} (z, u) - h_{1}, φ ⟩ + ⟨ f_{2} (z, u) - h_{2}, ψ ⟩ = 0, \end{matrix} \end{matrix}$ where θ fulfills the explicit representation (2.1). In addition, if a weak solution further satisfies $\begin{matrix} z \in C ([0, \infty); D (A)) \cap C^{1} ([0, \infty); H), \end{matrix}$ then it is called a strong solution.

We define the total energy of the solutions $U = (z, u, z_{t}, u_{t}, θ)$ of (2.3) by $\begin{matrix} (2.17) & E (t) : = E (t) + \int_{0}^{L} F (z (t), u (t)) d x - ⟨ h_{1}, z (t) ⟩ - ⟨ h_{2}, u (t) ⟩, \end{matrix}$ where $E (t)$ is the natural energy corresponding to the system (2.3): $\begin{matrix} (2.18) & E (t) = \frac{1}{2} {‖ (z (t), u (t), z_{t} (t), u_{t} (t), θ (t)) ‖}_{H}^{2} . \end{matrix}$

Now, we need the following auxiliary result to be used in the sequel.
Lemma 2.3.
Suppose that* $U = (z, u, z_{t}, u_{t}, θ)$ is a strong solution to ( 2.3 )–( 2.4 ). Then, the total energy ( 2.17 ) satisfies $\begin{matrix} (2.19) & \frac{d}{d t} E (t) ⩽ - ‖ z_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) ‖ u_{t} ‖_{2} - (\frac{1 - ϵ}{2}) {‖ θ (1) ‖}_{2}^{2} . \end{matrix}$ Moreover, there exist constants $β_{0}, C_{F} > 0$ such that $\begin{matrix} (2.20) & E (t) ⩾ β_{0} {‖ U (t) ‖}_{H}^{2} - C_{F}, \forall t ⩾ 0, \end{matrix}$ and $\begin{matrix} (2.21) & E (t) ⩽ C_{F} ({‖ U (t) ‖}_{H}^{q + 1} + 1), \forall t ⩾ 0 . \end{matrix}$
Proof.
Multiplying the first equation in (2.3) by $z_{t}$ , the second one by $u_{t}$ and the third one by θ, respectively, and using integration by parts, yields $\begin{matrix} \frac{d}{d t} E (t) = - ‖ z_{t} ‖_{2}^{2} - ‖ u_{t} ‖_{2}^{2} - ϵ ⟨ θ (1), u_{t} ⟩ + \frac{1}{2} (‖ u_{t} ‖_{2}^{2} - {‖ θ (1) ‖}_{2}^{2}) . \end{matrix}$ By Hölder’s and Young’s inequalities the following is obtained $\begin{matrix} \frac{d}{d t} E (t) ⩽ - ‖ z_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) ‖ u_{t} ‖_{2} - (\frac{1 - ϵ}{2}) {‖ θ (1) ‖}_{2}^{2} . \end{matrix}$ Thus (2.19) holds.

Now, from (2.13) and (2.8) we find $\begin{matrix} (2.22) & \begin{matrix} \int_{0}^{L} F (z, u) d x & ⩾ - η (‖ z ‖_{2}^{2} + ‖ u ‖_{2}^{2}) - L m_{F} \\ ⩾ - η η_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) - L m_{F} \\ ⩾ - η η_{0} {‖ U (t) ‖}_{H}^{2} - L m_{F} . \end{matrix} \end{matrix}$ By (2.22) we have $\begin{matrix} E (t) ⩾ (\frac{1}{2} - η η_{0}) {‖ U (t) ‖}_{H}^{2} - L m_{F} - ⟨ h_{1}, z (t) ⟩ - ⟨ h_{2}, u (t) ⟩ . \end{matrix}$ Now letting $\begin{matrix} (2.23) & β_{0} = \frac{1}{4} (1 - 2 η η_{0}) > 0, \end{matrix}$ and using the estimate $\begin{matrix} ⟨ h_{1}, z (t) ⟩ + ⟨ h_{2}, u (t) ⟩ ⩽ \frac{β_{0}}{η_{0}} (‖ z ‖_{2}^{2} + ‖ u ‖_{2}^{2}) + \frac{η_{0}}{4 β_{0}} (‖ h_{1} ‖_{2}^{2} + ‖ h_{2} ‖_{2}^{2}), \end{matrix}$ the inequality in (2.20) is obtained with $\begin{matrix} C_{F} = L m_{F} + \frac{η_{0}}{4 β_{0}} (‖ h_{1} ‖_{2}^{2} + ‖ h_{2} ‖_{2}^{2}) . \end{matrix}$ On the other hand, from the assumption (2.7) the embedding $H_{*}^{1} (0, L) ↪ L^{\infty} (0, L)$ we have $\begin{matrix} \int_{0}^{L} F (z, u) d x ⩽ C (‖ z_{x} ‖_{2}^{q + 1} + ‖ u_{x} ‖_{2}^{q + 1} + 1) . \end{matrix}$ Hence, using the last estimate and (2.12), we conclude that (2.21) holds. The proof is complete. □

Now we give the global well-posedness of problem (1.11)–(1.12). Theorem 2.4 (Well-posedness).

Suppose that Assumption 2.1 holds. Then for any initial data $U_{0} \in H$ , the system ( 1.11 )–( 1.12 ) has a unique weak solution $U = (z, u, z_{t}, u_{t}, θ)$ satisfying $\begin{matrix} U \in C ([0, \infty); H), U (0) = U_{0} . \end{matrix}$ If $U_{0} \in D (A)$ , then the solution is strong. Moreover, the weak solutions depend continuously on the initial data $U_{0}$ in $H$ .

Proof.
The proof of the theorem is divided into two steps.

Step 1: Local solutions. It is easy to check that operator $A$ given in (2.15) is a maximal monotone operator in $H$ and $F : H \to H$ is locally Lipschitz by assumption (2.7). Then, by [8, Theorem 7.2], for all $U_{0} \in D (A)$ there exists $t_{max} ⩽ \infty$ and a unique strong solution U for (2.14) defined on the interval $[0, t_{max})$ . Moreover, if $U_{0} \in H$ , (2.14) has a unique weak solution $U \in C ([0, t_{max}); H)$ and such solution satisfies ${lim sup}_{t \to t_{max}^{-}} ‖ U (t) ‖_{H} = \infty$ provided that $t_{max} < \infty$ .

Step 2: Global solutions. Let U be a strong solution defined in $[0, t_{max})$ . By (2.19), we have $\begin{matrix} (2.24) & E (t) ⩽ E (0), \forall t \in [0, t_{max}) . \end{matrix}$ Using (2.20) and (2.24) yields $\begin{matrix} (2.25) & {‖ U (t) ‖}_{H}^{2} ⩽ \frac{1}{β_{0}} (E (0) + C_{F}), \forall t \in [0, t_{max}) . \end{matrix}$ By a density argument, the inequality (2.25) also holds for weak solutions. Therefore $t_{max} = \infty$ .

Finally, let $U^{1}$ , $U^{2}$ be two weak solutions. By using standard computations in the difference of solutions, we conclude that there exists $C_{0} = C_{0} (U^{1} (0), U^{2} (0)) > 0$ such that $\begin{matrix} (2.26) & {‖ U^{1} (t) - U^{2} (t) ‖}_{H}^{2} ⩽ e^{C_{0} T} {‖ U^{1} (0) - U^{2} (0) ‖}_{H}^{2}, \forall t \in [0, T] . \end{matrix}$ This proves the continuous dependence. The proof is complete. □

3. A review on dynamical systems and global attractors

In this section, we will outline some concepts and results related to dynamical systems that will be important for this work; see [2, 5, 9, 20, 22, 31, 36]) and the references therein. In particular, the reference [9] is more relevant for the rest of the paper.

Definition 3.1.
A dynamical system is a pair $(H, S (t))$ where H is a Banach space and $S (t)$ is a strongly continuous semigroup in H.
Definition 3.2.
A compact set $A \subset H$ is called a global attractor for $(H, S (t))$ if $A$ is an invariant set, that is, $S (t) A = A$ , for all $t ⩾ 0$ and $A$ is uniformly attracting, that is, for every bounded set $B \subset H$ , we have $\begin{matrix} lim_{t \to + \infty} {dist}_{H} (S (t) B, A) = 0, \end{matrix}$ where ${dist}_{H}$ denotes the Hausdorff semi-distance.
Definition 3.3.
A dynamical system $(H, S (t))$ is called dissipative if it has an absorbing set, that is, a bounded set $B \subset H$ such that for any bounded set $D \subset H$ there exists $t_{D} > 0$ with $\begin{matrix} S (t) D \subset B, \forall t ⩾ t_{D} . \end{matrix}$
Definition 3.4.
A dynamical system $(H, S (t))$ is called asymptotically compact if for any bounded set $D \subset H$ and any sequences $0 < t_{n} \to \infty$ and $x_{n} \in D$ the set ${S (t_{n}) x_{n}}$ is precompact in H.
Definition 3.5.
Let $N$ be the sets of stationary points of $(H, S (t))$ , that is, $\begin{matrix} N : = {h \in H; S (t) h = h, \forall t ⩾ 0} . \end{matrix}$ Then the unstable manifold emanating from $N$ , represented by $M_{+}^{u} (N)$ , is the set of all $h \in H$ such that there is a full trajectory $γ = {u (t); t \in R}$ satisfying $\begin{matrix} (3.1) & u (0) = h and lim_{t \to - \infty} {dist}_{H} (u (t), N) = 0 . \end{matrix}$
Definition 3.6.
A dynamical system $(H, S (t))$ is called gradient, if there exists a strict Lyapunov function on H, that is, there exists a continuous function Φ such that $t \mapsto Φ (S (t) y)$ is non-increasing for any $y \in H$ , and if $Φ (S (t) y_{0}) = Φ (y_{0})$ for all $t > 0$ and some $y_{0} \in H$ , then $y_{0}$ is a stationary point of $(H, S (t))$ .
Theorem 3.7 ([9, Corollary 7.5.7]).

Let $(H, S (t))$ be an asymptotically compact gradient system on a Banach space H, with the corresponding Lyapunov functional denoted by Φ. Suppose that $\begin{matrix} Φ (y) \to \infty if and only if ‖ y ‖_{H} \to \infty, \end{matrix}$ and that the set of stationary points $N$ is bounded. Then the system $(H, S (t))$ possesses a compact global attractor characterized by $A = M_{+}^{u} (N)$ .

Definition 3.8.
The fractal dimension of a compact set M in H is defined by $\begin{matrix} (3.2) & {dim}_{f}^{H} M : = lim_{ε \to 0} sup \frac{ln n (M, ε)}{ln (1 / ε)}, \end{matrix}$ where $n (M, ε)$ is the minimal number of closed balls of radius ε which cover M.
Definition 3.9.
A compact set $A_{\exp} \subset H$ is called an exponential attractor for $(H, S (t))$ if
$A_{\exp}$ is a positively invariant set, that is, $S (t) A_{\exp} \subset A_{\exp}$ for all $t ⩾ 0$ ;

$A_{\exp}$ has finite fractal dimension in H;

$A_{\exp}$ attracts bounded sets of H at an exponential rate, that is, for any bounded set $D \subset H$ there exist $t_{D}, C_{D}, γ_{D} > 0$ such that $\begin{matrix} {dist}_{H} (S (t) D, A_{\exp}) ⩽ C_{D} e^{- γ_{D} (t - t_{D})}, \forall t ⩾ t_{D} . \end{matrix}$

Let X, Y and Z be reflexive Banach spaces with X compactly embedded in Y. We consider the space $H = X \times Y \times Z$ and the dynamical system $(H, S (t))$ given by $\begin{matrix} (3.3) & S (t) y = (u (t), u_{t} (t), θ (t)), y = (u (0), u_{t} (0), θ (0)) \in H, \end{matrix}$ where the function u has the regularity $\begin{matrix} u \in C ([0, \infty); X) \cap C^{1} ([0, \infty); Y), θ \in C ([0, \infty); Z) . \end{matrix}$

The dynamical system $(H, S (t))$ is called quasi-stable on a set $B \subset H$ if there exist a compact seminorm $n_{X} (\cdot)$ on the space X and nonnegative scalar functions $a (t)$ , $b (t)$ and $c (t)$ on $[0, \infty)$ such that

$a (t)$ and $c (t)$ are locally bounded on $[0, \infty)$ .

$b (t) \in L^{1} (0, \infty)$ possesses the property $\begin{matrix} lim_{t \to \infty} b (t) = 0 . \end{matrix}$

for every $y_{1}, y_{2} \in B$ and $t > 0$ the following relations $\begin{matrix} (3.4) & {‖ S (t) y_{1} - S (t) y_{2} ‖}_{H}^{2} ⩽ a (t) ‖ y_{1} - y_{2} ‖_{H}^{2} \end{matrix}$ and $\begin{matrix} (3.5) & {‖ S (t) y_{1} - S (t) y_{2} ‖}_{H}^{2} ⩽ b (t) ‖ y_{1} - y_{2} ‖_{H}^{2} + c (t) sup_{0 ⩽ s ⩽ t} {[n_{X} (u^{1} (s) - u^{2} (s))]}^{2} \end{matrix}$ hold. Here we denote $S (t) y_{j} = (u^{j} (t), u_{t}^{j} (t), θ^{j} (t))$ , $j = 1, 2$ .

The following results, which can be found in [9], show us how strong the property of quasi-stability is for a dynamical system. Theorem 3.10 ([9, Proposition 7.9.4]).

Let $(H, S (t))$ be a dynamical system satisfying ( 3.3 ). Assume that $(H, S (t))$ is quasi-stable over any bounded invariant set $B \subset H$ . Then, $(H, S (t))$ is asymptotically compact.

Theorem 3.11 ([9, Theorem 7.9.6]).

Suppose $(H, S (t))$ be a dynamical system satisfying ( 3.3 ). Assume that $(H, S (t))$ possesses a compact global attractor $A$ and is quasi-stable on $A$ . Then the fractal dimension of $A$ is finite.

Theorem 3.12 ([9, Theorem 7.9.9]).

Let $(H, S (t))$ be a dynamical system satisfying ( 3.3 ). Assume that $(H, S (t))$ is dissipative and quasi-stable on some bounded absorbing set B. Assume also that there exists an extended space $\tilde{H} \supseteq H$ such that $\begin{matrix} {‖ S (t_{1}) y - S (t_{2}) y ‖}_{\tilde{H}} ⩽ C_{B, T} | t_{1} - t_{2} |^{γ}, \forall t_{1}, t_{2} \in [0, T], \end{matrix}$ where $C_{B, T} > 0$ and $γ \in [0, 1)$ are constants. Then the dynamical system possesses a generalized exponential attractor $A_{\exp} \subset H$ whose dimension is finite in the space $\tilde{H}$ .

Theorem 3.13 ([9, Theorem 7.9.8]).

Let $(H, S (t))$ be a dynamical system satisfying ( 3.3 ). Assume that the dynamical system possesses a compact global attractor $A$ and is quasi-stable on the attractor $A$ . Moreover, we assume that (QS3) holds for the function $c (t)$ possessing the property $c_{\infty} = {sup}_{t \in [0, \infty)} c (t) < \infty$ . Then any full trajectory $(u (t), u_{t} (t), z (t))$ in the global attractor has the following regularity properties, $\begin{matrix} u \in L^{\infty} (R; X) \cap C (R; Y), u_{t t} \in L^{\infty} (R; Y), z_{t} \in L^{\infty} (R, Z) . \end{matrix}$ Moreover, there exists $R > 0$ such that $\begin{matrix} {‖ u_{t} (t) ‖}_{X}^{2} + {‖ u_{t t} (t) ‖}_{Y}^{2} + {‖ z_{t} (t) ‖}_{Z}^{2} ⩽ R^{2}, \forall t \in R, \end{matrix}$ where R depends on the constant $c_{\infty}$ , seminorm $n_{X}$ , and also embedding properties of X into Y.

4. Main results

First, we observe that the system (2.4)–(2.4) defines a dynamical system $(H, S_{ϵ} (t))$ , where $H$ is given in (2.11), and $S_{ϵ} (t) : H \to H$ is the strongly continuous semigroup given by $\begin{matrix} S_{ϵ} (t) U_{0} = (z (t), u (t), z_{t} (t), u_{t} (t), θ (t)), t ⩾ 0, \end{matrix}$ with $(z (t), u (t), z_{t} (t), u_{t} (t), θ (t))$ being the weak solution of (2.4)–(2.4) with initial data $U_{0} \in H$ .

The main result for long-time dynamics is given by the following theorem whose proof will be provided in the next section.

Theorem 4.1.
Suppose that Assumption 2.1 holds. Then for each $ϵ \in [0, 1)$ , we have
The dynamical system $(H, S_{ϵ} (t))$ is quasi-stable on any bounded positively invariant set $B \subset H$ .

The dynamical system $(H, S_{ϵ} (t))$ possesses a unique compact global attractor $A_{ϵ} \subset H$ , which is characterized by the unstable manifold $A_{ϵ} = M_{+}^{u} (N)$ of the set of stationary solutions $\begin{matrix} N = \{(z, u, 0, 0, 0) \in H | \begin{matrix} - a_{1} z_{x x} - a_{2} u_{x x} + f_{1} (z, u) = h_{1} \\ - a_{3} u_{x x} - a_{2} z_{x x} + f_{2} (z, u) = h_{2} \end{matrix}\} . \end{matrix}$

The attractor $A_{ϵ}$ has finite fractal and Hausdorff dimension ${dim}_{H}^{f} A_{ϵ}$ .

The complete trajectories $(z, u, z_{t}, u_{t}, θ)$ in $A_{ϵ}$ has further regularity $\begin{matrix} (4.1) & {‖ (z_{t}, u_{t}, z_{t t}, u_{t t}, θ_{t}) ‖}_{H}^{2} ⩽ R, \end{matrix}$ for some constant $R > 0$ depending on $A_{ϵ}$ .

The dynamical system $(H, S (t))$ has a generalized exponential attractor $A_{exp, ϵ}$ with finite fractal dimension in the extended space $\begin{matrix} H_{- 1} : = L^{2} (0, L) \times L^{2} (0, L) \times {(H_{}^{1} (0, L))}^{} \times {(H_{}^{1} (0, L))}^{} \times L^{2} ((0, L) \times (0, 1)), \end{matrix}$ where ${(H_{}^{1} (0, L))}^{}$ denotes the dual space of $H_{}^{1} (0, L)$ . In addition, from interpolation theorem, there exists a generalized exponential attractor whose fractal dimension is finite in a smaller extended space* $H_{- σ}$ , where $\begin{matrix} H = H_{0} \subset H_{- σ} \subset H_{- 1}, 0 < σ ⩽ 1 . \end{matrix}$

5. Proofs

5.1. Stabilizability inequality

The following lemma plays an important role to prove the existence of a global attractor and its properties. We usually call it the stabilizability inequality.

Lemma 5.1.
Suppose that Assumption 2.1 holds. Let B be a bounded positively invariant set in $H$ and let $S_{ϵ} (t) U^{j} = (z^{j}, u^{j}, z_{t}^{j}, u_{t}^{j}, θ^{j})$ be the weak solutions of ( 2.3 )–( 2.4 ) with initial condition $U^{j} \in B$ , $j = 1, 2$ . Then, there exist constants $μ_{0}, γ_{0}, C_{B}^{'} > 0$ such that $\begin{matrix} (5.1) & E (t) ⩽ γ_{0} e^{- μ_{0} t} E (0) + C_{B}^{'} \int_{0}^{t} e^{- μ_{0} (t - s)} ({‖ z (s) ‖}_{2 q}^{2} + {‖ u (s) ‖}_{2 q}^{2}) d s, \end{matrix}$ where $z = z^{1} - z^{2}$ and $u = u^{1} - u^{2}$ .
Proof.
To simplify the notation $z = z^{1} - z^{2}$ , $u = u^{1} - u^{2}$ and $θ = θ^{1} - θ^{2}$ , the following notation is adopted $\begin{matrix} F_{j} (z, u) = f_{j} (z^{1}, u^{1}) - f_{j} (z^{2}, u^{2}), j = 1, 2 . \end{matrix}$ Then, $(z, u, u_{t}, z_{t}, θ)$ solves the system $\begin{matrix} (5.2) & \{\begin{matrix} ρ_{z} z_{t t} - a_{1} z_{x x} - a_{2} u_{x x} + z_{t} + F_{1} (z, u) = 0, & in (0, L), \times (0, T), \\ ρ_{u} u_{t t} - a_{3} u_{x x} - a_{2} z_{x x} + u_{t} + ϵ θ (1) + F_{2} (z, u) = 0, & in (0, L) \times (0, T), \\ τ θ_{t} + θ_{y} = 0, & in (0, L) \times (0, 1) \times (0, T), \end{matrix} \end{matrix}$ with boundary conditions $\begin{matrix} z (0) = z_{x} (L) = u (0) = u_{x} (L) = 0, \end{matrix}$ and initial conditions $\begin{matrix} (5.3) & (z (0), u (0), z_{t} (0), u_{t} (0), θ (0)) = U^{1} - U^{2} . \end{matrix}$ Step 1. We claim that there exists a constant $C_{B} > 0$ such that $\begin{matrix} (5.4) & \begin{matrix} \frac{d}{d t} E (t) & ⩽ - \frac{1}{2} ‖ z_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{4}) ‖ u_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) {‖ θ (1) ‖}_{2}^{2}, \\ + C_{B} (‖ z ‖_{2 q} + ‖ u ‖_{2 q}) . \end{matrix} \end{matrix}$ Multiplying the first equation in (5.2) by $z_{t}$ , the second by $u_{t}$ , the third by θ, and integrating over $[0, L]$ , we obtain $\begin{matrix} (5.5) & \begin{matrix} \frac{d}{d t} E (t) & = - ‖ z_{t} ‖_{2}^{2} - ‖ u_{t} ‖_{2}^{2} - ϵ ⟨ θ (1), u_{t} ⟩ + \frac{1}{2} (‖ u_{t} ‖_{2}^{2} - {‖ θ (1) ‖}_{2}^{2}) \\ - ⟨ F_{1} (z, u), z_{t} ⟩ - ⟨ F_{2} (z, u), u_{t} ⟩ . \end{matrix} \end{matrix}$ By Hölder’s and Young’s inequalities the following is obtained $\begin{matrix} (5.6) & \begin{matrix} \frac{d}{d t} E (t) & ⩽ - ‖ z_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) ‖ u_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) {‖ θ (1) ‖}_{2}^{2} \\ - ⟨ F_{1} (z, u), z_{t} ⟩ - ⟨ F_{2} (z, u), u_{t} ⟩ . \end{matrix} \end{matrix}$ By using (2.7), we can deduce that $\begin{matrix} (5.7) & \begin{matrix} | ⟨ F_{1} (z, u), z_{t} ⟩ | \\ ⩽ C ({‖ z^{1} ‖}_{2 q}^{q - 1} + {‖ z^{2} ‖}_{2 q}^{q - 1} + {‖ u^{1} ‖}_{2 q}^{q - 1} + {‖ u^{2} ‖}_{2 q}^{q - 1} + 1) (‖ z ‖_{2 q} + ‖ u ‖_{2 q}) ‖ z_{t} ‖_{2} \\ ⩽ C_{B} (‖ z ‖_{2 q} + ‖ u ‖_{2 q}) ‖ z_{t} ‖_{2} ⩽ C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) + \frac{1}{2} ‖ z_{t} ‖_{2}^{2} . \end{matrix} \end{matrix}$ Analogously, we obtain $\begin{matrix} (5.8) & | ⟨ F_{2} (z, u), u_{t} ⟩ | ⩽ C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) + (\frac{1 - ϵ}{4}) ‖ u_{t} ‖_{2}^{2} . \end{matrix}$ Substituting the estimates (5.7) and (5.8) in (5.6) we conclude that (5.4) holds.

Step 2. Consider the functional $\begin{matrix} (5.9) & I (t) : = ρ_{z} ⟨ z_{t}, z ⟩ + ρ_{u} ⟨ u_{t}, u ⟩ + \frac{1}{2} ‖ z ‖_{2}^{2} + \frac{1}{2} ‖ u ‖_{2}^{2} . \end{matrix}$ By using ${(5.2)}_{1}$ and ${(5.2)}_{2}$ , we obtain $\begin{matrix} (5.10) & \begin{matrix} \frac{d}{d t} I (t) = & ρ_{z} ‖ z_{t} ‖_{2}^{2} + ρ_{u} ‖ u_{t} ‖_{2}^{2} - χ ‖ u_{x} ‖_{2}^{2} - {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2} \\ - ϵ ⟨ θ (1), u ⟩ - ⟨ F_{1} (z, u), z ⟩ - ⟨ F_{2} (z, u), u ⟩ . \end{matrix} \end{matrix}$ By Poincare’s and Young’s inequalities, we find $\begin{matrix} (5.11) & ϵ | ⟨ θ (1), u ⟩ | ⩽ C {‖ θ (1) ‖}_{2}^{2} + \frac{χ}{2} ‖ u_{x} ‖_{2}^{2} . \end{matrix}$ Similarly to (5.7) and (5.8), we have $\begin{matrix} (5.12) & | ⟨ F_{1} (z, u), z ⟩ | ⩽ C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}), | ⟨ F_{2} (z, u), u ⟩ | ⩽ C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) . \end{matrix}$ Inserting the estimates (5.11) and (5.12) into (5.10) yields $\begin{matrix} (5.13) & \begin{matrix} \frac{d}{d t} I (t) & ⩽ ρ_{z} ‖ z_{t} ‖_{2}^{2} + ρ_{u} ‖ u_{t} ‖_{2}^{2} - \frac{χ}{2} ‖ u_{x} ‖_{2}^{2} - {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2} \\ + C {‖ θ (1) ‖}_{2}^{2} + C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) . \end{matrix} \end{matrix}$ Step 3. We define of functional $P$ by $\begin{matrix} (5.14) & P (t) = τ \int_{0}^{L} \int_{0}^{1} e^{- 2 τ y} | θ |^{2} d y d x . \end{matrix}$ Deriving $P$ with respect to t and using the equation ${(5.2)}_{3}$ , we have $\begin{matrix} (5.15) & \begin{matrix} \frac{d}{d t} P (t) & = 2 τ \int_{0}^{L} \int_{0}^{1} e^{- 2 τ y} θ θ_{t} d y d x = - \int_{0}^{L} \int_{0}^{1} e^{- 2 τ y} \frac{\partial}{\partial y} | θ |^{2} d y d x \\ = ‖ u_{t} ‖_{2}^{2} - e^{- 2 τ} {‖ θ (1) ‖}_{2}^{2} - 2 τ \int_{0}^{L} \int_{0}^{1} e^{- 2 τ y} | θ |^{2} d y d x . \end{matrix} \end{matrix}$ Step 4. Consider the functional $\begin{matrix} L (t) : = N_{1} E (t) + I (t) + P (t) . \end{matrix}$ By Young’s inequality and (2.13), the following holds true $\begin{matrix} (5.16) & | L (t) - N_{1} E (t) | ⩽ | I (t) | + N_{2} | P (t) | ⩽ \tilde{η} E (t), \tilde{η} > 0 . \end{matrix}$ Consequently, for $N_{1} > \tilde{η}$ , $\begin{matrix} (5.17) & (N_{1} - \tilde{η}) E (t) ⩽ L (t) ⩽ (N_{1} + \tilde{η}) E (t), \forall t ⩾ 0 . \end{matrix}$ Step 5. By estimates (5.4), (5.13) and (5.15), we have $\begin{matrix} (5.18) & \begin{matrix} \frac{d}{d t} L (t) & ⩽ - [\frac{N_{1}}{2} - ρ_{z}] ‖ z_{t} ‖_{2}^{2} - [N_{1} (\frac{1 - ϵ}{4}) - ρ_{u} - 1] ‖ u_{t} ‖_{2}^{2} \\ - \frac{χ}{2} ‖ u_{x} ‖_{2}^{2} - {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2} - 2 τ e^{- 2 τ} \int_{0}^{L} \int_{0}^{1} | θ |^{2} d y d x \\ - [N_{1} (\frac{1 - ϵ}{2}) - C + e^{- 2 τ}] {‖ θ (1) ‖}_{2}^{2} \\ + (N_{1} + 1) C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) . \end{matrix} \end{matrix}$ Choosing $N_{1}$ large enough, we conclude that there exist constants $N_{0}, C_{B} > 0$ such that $\begin{matrix} (5.19) & \frac{d}{d t} L (t) ⩽ - N_{0} E (t) + C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}), \forall t ⩾ 0 . \end{matrix}$ Utilizing the second inequality in (5.17), we obtain $\begin{matrix} (5.20) & \frac{d}{d t} L (t) ⩽ - \frac{N_{0}}{N_{1} + \tilde{η}} L (t) + C_{B} (‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}) . \end{matrix}$ Now, the Gronwall’s Lemma is applied to (5.20) to get $\begin{matrix} L (t) ⩽ e^{- \frac{N_{0}}{N_{1} + \tilde{η}} t} L (0) + C_{B} \int_{0}^{t} e^{- \frac{N_{0}}{N_{1} + \tilde{η}} (t - s)} ({‖ z (s) ‖}_{2 q}^{2} + {‖ u (s) ‖}_{2 q}^{2}) d s . \end{matrix}$ Choosing $γ_{0} : = \frac{N_{1} + \tilde{η}}{N_{1} - \tilde{η}} > 0$ , $μ_{0} : = \frac{N_{0}}{N_{1} + \tilde{η}} > 0$ , $C_{B}^{'} : = \frac{C_{B}}{N_{1} - \tilde{η}} > 0$ and reusing (5.17), the following inequality holds $\begin{matrix} E (t) ⩽ γ_{0} e^{- μ_{0} t} E (0) + C_{B}^{'} \int_{0}^{t} e^{- μ_{0} (t - s)} ({‖ z (s) ‖}_{2 q}^{2} + {‖ u (s) ‖}_{2 q}^{2}) d s . \end{matrix}$ Therefore (5.1) is obtained. □

5.2. Gradient system and stationary points

Lemma 5.2.
Suppose that Assumption 2.1 holds. Then the dynamical system $(H, S_{ϵ} (t))$ is gradient.
Proof.
Let us define the function $Φ : H \to R$ by $\begin{matrix} (5.21) & Φ (S_{ϵ} (t) U) = \frac{1}{2} {‖ (z (t), u (t), z_{t} (t), u_{t} (t), θ (t)) ‖}_{H}^{2} + \int_{0}^{L} F (z (t), u (t)) d x . \end{matrix}$ If $U = (z_{0}, u_{0}, z_{1}, u_{1}, θ_{0}) \in H$ , then by (2.19) we obtain that $\begin{matrix} (5.22) & \frac{d}{d t} Φ (S_{ϵ} (t) U) = - ‖ z_{t} ‖_{2}^{2} - (\frac{1 - ϵ}{2}) ‖ u_{t} ‖_{2} - (\frac{1 - ϵ}{2}) {‖ θ (1) ‖}_{2}^{2} ⩽ 0 . \end{matrix}$ Then, $t \mapsto Φ (S_{ϵ} (t) U)$ is a non-increasing function.

Suppose that $Φ (S_{ϵ} (t) U) = Φ (U)$ for all $t ⩾ 0$ . Then, by (5.22) it is easy see that $\begin{matrix} (5.23) & z_{t} = u_{t} = 0 a.e. in (0, L), \forall t ⩾ 0 . \end{matrix}$ Thus, $z (t) = z_{0}$ and $u (t) = u_{0}$ for all $t ⩾ 0$ . By (5.23) and (2.1), we conclude that $θ = 0$ . Then $U = (z_{0}, u_{0}, 0, 0, 0)$ is a stationary point of $(H, S_{ϵ} (t))$ . This proves that Φ is a strict Lyapunov function. □
Lemma 5.3.
Suppose that Assumption 2.1 holds. Then the set of equilibrium points $N$ of $(H, S_{ϵ} (t))$ is bounded in $H$ .
Proof.
Let $U \in N$ . We know that $U = (z, u, 0, 0, 0)$ , and it satisfies the elliptic equations $\begin{matrix} (5.24) & \{\begin{matrix} - a_{1} z_{x x} - a_{2} u_{x x} + f_{1} (z, u) = h_{1}, \\ - a_{3} u_{x x} - a_{2} z_{x x} + f_{2} (z, u) = h_{2} . \end{matrix} \end{matrix}$ Multiplying the first equation in (5.24) by z, the second by u integrating the result over $(0, L)$ , we obtain $\begin{matrix} (5.25) & χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2} = - ⟨ f_{1} (z, u), z ⟩ - ⟨ f_{2} (z, u), u ⟩ + ⟨ h_{1}, z ⟩ + ⟨ h_{2}, u ⟩ . \end{matrix}$ Hence, using (2.6), (2.8) and (2.9), we get $\begin{matrix} (5.26) & - ⟨ f_{1} (z, u), z ⟩ - ⟨ f_{2} (z, u), u ⟩ ⩽ 2 η η_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) + 2 L m_{F} . \end{matrix}$ Therefore, from (5.25), (5.26) and (2.23), we obtain that $\begin{matrix} (5.27) & 4 β_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) ⩽ 2 L m_{F} + ⟨ h_{1}, z ⟩ + ⟨ h_{2}, u ⟩ . \end{matrix}$ By Young’s inequality and (2.13), we have $\begin{matrix} ⟨ h_{1}, z ⟩ + ⟨ h_{2}, u ⟩ & ⩽ \frac{β_{0}}{η_{0}} (‖ z ‖_{2}^{2} + ‖ u ‖_{2}^{2}) + \frac{η_{0}}{4 β_{0}} (‖ h_{1} ‖_{2}^{2} + ‖ h_{2} ‖_{2}^{2}) \\ ⩽ β_{0} (χ ‖ u_{x} ‖_{2}^{2} + {‖ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x} ‖}_{2}^{2}) + \frac{η_{0}}{4 β_{0}} (‖ h_{1} ‖_{2}^{2} + ‖ h_{2} ‖_{2}^{2}) . \end{matrix}$ Inserting the last estimate into (5.27), we obtain $\begin{matrix} (5.28) & 3 β_{0} ‖ U ‖^{2} ⩽ 2 L m_{F} + \frac{η_{0}}{4 β_{0}} (‖ h_{1} ‖_{2}^{2} + ‖ h_{2} ‖_{2}^{2}) \end{matrix}$ which shows that the set $N$ is bounded in $H$ . □
5.3. Proof of Theorem 4.1

(i) We consider a bounded positively invariant set $B \subset H$ with respect to $S_{ϵ} (t)$ , denote $S_{ϵ} (t) U^{j} = (z^{j}, u^{j}, z_{t}^{j}, u_{t}^{j}, θ^{j})$ for $U^{j} \in B$ , $j = 1, 2$ , and set $z = z^{1} - z^{2}$ , $u = u^{1} - u^{2}$ as before. It follows from (2.26) that $\begin{matrix} {‖ S_{ϵ} (t) U^{1} - S_{ϵ} (t) U^{2} ‖}_{H}^{2} ⩽ a (t) {‖ U^{1} - U^{2} ‖}_{H}^{2}, \forall t ⩾ 0, \end{matrix}$ where $a (t) = e^{C_{0} T}$ , and therefore, (3.4) holds. Denote by $X = {(H_{*}^{1} (0, L))}^{2}$ , $Y = {(L^{2} (0, L))}^{2}$ and $Z = L^{2} ((0, L) \times (0, 1))$ . Thus, $H = X \times Y \times Z$ and Lemma 5.1 imply that $\begin{matrix} {‖ S_{ϵ} (t) U^{1} - S_{ϵ} (t) U^{2} ‖}_{H}^{2} ⩽ b (t) {‖ U^{1} - U^{2} ‖}_{H}^{2} + c (t) sup_{0 ⩽ s ⩽ t} {[n_{X} (z (s), u (s))]}^{2}, \forall U^{1}, U^{2} \in B, \end{matrix}$ where $\begin{matrix} b (t) = γ_{0} e^{- μ_{0} t}, c (t) = C_{B}^{'} \int_{0}^{t} e^{- μ_{0} (t - s)} d s, \end{matrix}$ and $n_{X} (\cdot)$ is the compact seminorm in X defined by $\begin{matrix} n_{X} (z, u) = \sqrt{‖ z ‖_{2 q}^{2} + ‖ u ‖_{2 q}^{2}}, (z, u) \in X . \end{matrix}$ It is easy to get that $\begin{matrix} b (t) \in L^{1} (0, \infty) and lim_{t \to \infty} b (t) = 0 . \end{matrix}$ Since $B \subset H$ is bounded, we know that $c (t)$ is locally bounded on $[0, \infty)$ . Then, the dynamics system $(H, S_{ϵ} (t))$ is quasi-stable on any bounded positively invariant set $B \subset H$ .

(ii) Since the system $(H, S_{ϵ} (t))$ is quasi-stable, Theorem 3.10 indicates that $(H, S_{ϵ} (t))$ is asymptotically compact. In addition, Lemmas 5.2 and 5.3 indicate that $(H, S_{ϵ} (t))$ is also gradient and the set of its stationary points $N$ is bounded. In addition, estimates (2.20) and (2.21) imply $\begin{matrix} Φ (U) \to \infty if and only if ‖ U ‖_{H} \to \infty . \end{matrix}$ Therefore all the assumptions of Theorem 3.7 are fulfilled and consequently the system $(H, S_{ϵ} (t))$ has a global attractor $A_{ϵ} = M_{+}^{u} (N)$ .

(iii) From the above, the dynamical system $(H, S_{ϵ} (t))$ is quasi-stable on the global attractor $A_{ϵ}$ , then we can apply Theorem 3.11 to conclude that the global attractor $A_{ϵ}$ has finite fractal dimension.

(iv) Since we have shown that $(H, S_{ϵ} (t))$ is quasi-stable on the global attractor $A_{ϵ}$ with $c_{\infty} = {sup}_{t \in R^{+}} c (t) = C_{A_{ϵ}} < \infty$ , it follows from Theorem 3.13 that any complete trajectory $(z, u, z_{t}, u_{t}, θ)$ in $A_{ϵ}$ enjoys the following regularity properties $\begin{array}{l} z_{t}, u_{t} \in L^{\infty} (R; H_{*}^{1} (0, L)) \cap C (R; L^{2} (0, L)), z_{t t}, u_{t t} \in L^{\infty} (R; L^{2} (0, L)), \\ θ_{t} \in L^{\infty} (R; L^{2} ((0, L) \times (0, 1))) . \end{array}$ Therefore, (4.1) holds.

(v) Now we take $B = {z : Φ (z) ⩽ R}$ , where Φ is the strict Lyapunov functional defined in (5.21). Then we know that the set B is a positively invariant absorbing set for R large enough. Hence the system $(H, S_{ϵ} (t))$ is quasi-stable on B. Then, for any $U \in B$ there exists $C_{B} > 0$ such that $\begin{matrix} (5.29) & {‖ S_{ϵ} (t) U ‖}_{H} ⩽ C_{B}, \forall t \in [0, T] . \end{matrix}$ Using (5.29) in (1.11) and the fact that nonlinear terms are locally Lipschitz continuous on $H$ , the following holds true $\begin{matrix} {‖ \frac{d}{d t} S_{ϵ} (t) U ‖}_{H_{- 1}} ⩽ C_{B}, \forall t \in [0, T] . \end{matrix}$ Hence, $\begin{matrix} (5.30) & {‖ S_{ϵ} (t_{2}) U - S_{ϵ} (t_{2}) U ‖}_{H_{- 1}} ⩽ | \int_{t_{1}}^{t_{2}} {‖ \frac{d}{d t} S_{ϵ} (t) U ‖}_{H_{- 1}} d t | ⩽ C_{B} | t_{1} - t_{2} |, \forall t_{1}, t_{2} \in [0, T] . \end{matrix}$ This proves that $t \mapsto S_{ϵ} (t) U$ is Hölder continuous in $H_{- 1}$ for any $U \in B$ with exponent $σ = 1$ . Then from Theorem 3.12 we conclude that $(S_{ϵ} (t), H)$ has a generalized exponential attractor $A_{exp, ϵ}$ whose fractal dimension is finite in $H_{- 1}$ . Following the same arguments in [9, 23], we can obtain the existence of exponential attractors in $H_{- σ}$ with $σ \in (0, 1)$ . The proof is complete.

6. Upper semicontinuity

In this section, we investigate the upper semicontinuity of global attractors $A_{ϵ}$ with respect to the parameter $ϵ \to 0$ . Observe that the existence of global attractor $A_{0}$ for the limit problem $(ϵ = 0)$ is proved in [33]. Firstly, we prove the existence of an absorbing set independently of $ϵ \in [0, 1)$ .

The following lemma will be used to obtain a uniform (with respect to the parameter ϵ of the problem) bound for the attractor (see [9, Remark 7.5.8]).

Lemma 6.1.

Under the assumptions of Lemma 5.2 , the following relation is valid: $\begin{matrix} sup_{U \in A_{ϵ}} Φ (U) ⩽ sup_{U \in N} Φ (U) . \end{matrix}$

Lemma 6.2.

The dynamical system $(H, S_{ϵ} (t))$ has a bounded absorbing set $B$ independent of $ϵ \in [0, 1)$ .

Proof.

Let Φ be the Lyapunov functional given in (5.21). By (2.20), (2.21) and Lemma 6.1, we obtain $\begin{matrix} sup_{U \in A_{ϵ}} ‖ U ‖_{H}^{2} & ⩽ \frac{{sup}_{U \in A_{ϵ}} Φ (U) + C_{1}}{C_{0}} \\ ⩽ \frac{{sup}_{U \in N} Φ (U) + C_{1}}{C_{0}} \\ ⩽ \frac{C_{2} (1 + {sup}_{U \in N} ‖ U ‖_{H}^{q + 1}) + C_{1}}{C_{0}} . \end{matrix}$ Hence, by (5.28), we conclude that there exists a constant $R_{0} > 0$ independent of ϵ such that $\begin{matrix} sup_{z \in A_{ϵ}} ‖ z ‖_{H}^{2} ⩽ R_{0}^{2} . \end{matrix}$ Therefore, the closed ball $B = B (0, R_{0} + 1) = {z \in H : ‖ z ‖_{H} ⩽ R_{0} + 1}$ is a bounded absorbing independent of $ϵ \in [0, 1)$ . □

Lemma 6.3.

Let B be a bounded subset in $H$ and $U^{ϵ} = (z_{0}^{ϵ}, u_{0}^{ϵ}, z_{1}^{ϵ}, u_{1}^{ϵ}, θ^{ϵ}) \in B$ initial conditions associated to problem with corresponding solutions $S_{ϵ} (t) U^{ϵ} = (z^{ϵ} (t), u^{ϵ} (t), z_{t}^{ϵ} (t), u_{t}^{ϵ} (t), θ^{ϵ} (t))$ . Then there exists a constant $K_{B} > 0$ independent of ϵ such that $\begin{matrix} E_{ϵ} (t) ⩽ K_{B} and {‖ θ^{ϵ} (1, t) ‖}_{2} + {‖ S_{ϵ} (t) U^{ϵ} ‖}_{H} ⩽ K_{B}, \forall ϵ \in [0, 1), t ⩾ 0 . \end{matrix}$

Proof.

Using (2.20) and (2.21), we obtain $\begin{matrix} β_{0} {‖ S_{ϵ} (t) U^{ϵ} ‖}_{H}^{2} - C_{F} ⩽ E_{ϵ} (t) ⩽ E_{ϵ} (0) ⩽ C_{F} ({‖ U^{ϵ} ‖}_{H}^{q + 1} + 1) ⩽ C_{B} . \end{matrix}$ By the relation (2.1), we find that $\begin{matrix} {‖ θ^{ϵ} (1, t) ‖}_{2}^{2} = {‖ z_{t}^{ϵ} (t - τ) ‖}_{2}^{2} ⩽ C {‖ S_{ϵ} (t - τ) U^{ϵ} ‖}_{H}^{2} ⩽ C sup_{t ⩾ 0} {‖ S_{ϵ} (t) U^{ϵ} ‖}_{H}^{2} ⩽ C (\frac{C_{B} + C_{F}}{β_{0}}) . \end{matrix}$ The proof is complete. □

Theorem 6.4.

Suppose that Assumption 2.1 holds. Let $ϵ_{n} \to 0$ be a sequence of positive numbers and $U^{n} = (z^{n}, u^{n}, z_{t}^{n}, u_{t}^{n}, θ^{n})$ the weak solution to ( 2.3 )–( 2.4 ) with $ϵ = ϵ_{n}$ and initial condition $U_{0} = (z_{0}, u_{0}, z_{1}, u_{1}, f_{0}) \in H$ . Then, there exists a weak solution $U = (z, u, z_{t}, u_{t}, θ)$ of ( 2.3 )–( 2.4 ) with $ϵ = 0$ and same initial data such that for any $T > 0$ $\begin{matrix} (6.1) & \{\begin{matrix} (z^{n}, u^{n}) \to (z, u) & weakly star in L^{\infty} (0, T; {(H_{*}^{1} (0, L))}^{2}), \\ (z_{t}^{n}, u_{t}^{n}) \to (z_{t}, u_{t}) & weakly star in L^{\infty} (0, T; {(L^{2} (0, L))}^{2}), \\ θ^{n} \to θ & weakly star in L^{\infty} (0, T; L^{2} ((0, L) \times (0, 1)) . \end{matrix} \end{matrix}$

Proof.

We consider the subset $B = {U_{0}}$ . By Lemma 6.3 there exists a constant $K > 0$ independent of n such that $\begin{matrix} (6.2) & {‖ θ^{n} (1, t) ‖}_{2} + {‖ U^{n} (t) ‖}_{H} ⩽ K, \forall t ⩾ 0 . \end{matrix}$ Thus, for any $T > 0$ , $\begin{array}{l} (6.3) & (z^{n}, u^{n}) \to (z, u) weakly star in L^{\infty} (0, T; {(H_{*}^{1} (0, L))}^{2}), \\ (6.4) & (z_{t}^{n}, u_{t}^{n}) \to (z_{t}, u_{t}) weakly star in L^{\infty} (0, T; {(L^{2} (0, L))}^{2}), \\ (6.5) & θ^{n} \to θ weakly star in L^{\infty} (0, T; L^{2} ((0, L) \times (0, 1))) . \end{array}$ By definition of weak solution, we have $\begin{matrix} (6.6) & \begin{matrix} ρ_{z} \frac{d}{d t} ⟨ z_{t}^{n}, φ ⟩ + ρ_{u} \frac{d}{d t} ⟨ u_{t}^{n}, ψ ⟩ + χ ⟨ u_{x}^{n}, ψ_{x} ⟩ + ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x}^{n} + \sqrt{a_{1}} z_{x}^{n}, \frac{a_{2}}{\sqrt{a_{1}}} ψ_{x} + \sqrt{a_{1}} φ_{x} ⟩ \\ + ⟨ z_{t}^{n}, φ ⟩ + ⟨ u_{t}^{n}, ψ ⟩ + ϵ_{n} ⟨ θ^{n} (1, t), ψ ⟩ + ⟨ f_{1} (z^{n}, u^{n}) - h_{1}, φ ⟩ + ⟨ f_{2} (z^{n}, u^{n}) - h_{2}, ψ ⟩ = 0, \end{matrix} \end{matrix}$ for all $φ, ψ \in H_{*}^{1} (0, L)$ , where $\begin{matrix} (6.7) & θ^{n} (x, y, t) = u_{t}^{n} (x, t - τ y) in (0, L) \times (0, 1) \times (0, T) . \end{matrix}$ Using the convergences in (6.3)–(6.4) it is easy see that $\begin{matrix} lim_{n \to \infty} ⟨ z_{t}^{n}, φ ⟩ = ⟨ z_{t}, φ ⟩, lim_{n \to \infty} ⟨ u_{t}^{n}, ψ ⟩ = ⟨ u_{t}, ψ ⟩, lim_{n \to \infty} ⟨ u_{x}^{n}, ψ_{x} ⟩ = ⟨ u_{x}, ψ_{x} ⟩, \\ lim_{n \to \infty} ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x}^{n} + \sqrt{a_{1}} z_{x}^{n}, \frac{a_{2}}{\sqrt{a_{1}}} ψ_{x} + \sqrt{a_{1}} φ_{x} ⟩ = ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x}, \frac{a_{2}}{\sqrt{a_{1}}} ψ_{x} + \sqrt{a_{1}} φ_{x} ⟩ . \end{matrix}$ Using (2.7) and the embedding $H_{*}^{1} (0, L) ↪ L^{\infty} (0, L)$ we deduce that $\begin{matrix} ⟨ f_{1} (z^{n}, u^{n}) - f_{1} (z, u), φ ⟩ & = \int_{0}^{L} (f_{1} (z^{n}, u^{n}) - f_{1} (z, u)) φ d x \\ ⩽ C ({‖ U^{n} (t) ‖}_{H}^{q - 1} + {‖ U (t) ‖}_{H}^{q - 1} + 1) ({‖ z^{n} - z ‖}_{2} + {‖ u^{n} - u ‖}_{2}) ‖ φ ‖_{2} \\ ⩽ C (K^{q - 1} + {‖ U (t) ‖}_{H}^{q - 1} + 1) ‖ φ ‖_{2} ({‖ z^{n} - z ‖}_{2} + {‖ u^{n} - u ‖}_{2}) . \end{matrix}$ By (6.3), (6.4) and Simon’s compactness Theorem [35] we have $\begin{matrix} (z^{n}, u^{n}) \to (z, u) strongly in C (0, T; {(L^{2} (0, L))}^{2}) . \end{matrix}$ Consequently $\begin{matrix} lim_{n \to \infty} ⟨ f_{1} (z^{n}, u^{n}), φ ⟩ = ⟨ f_{1} (z, u), φ ⟩ . \end{matrix}$ Analogously, we have $\begin{matrix} lim_{n \to \infty} ⟨ f_{2} (z^{n}, u^{n}), ψ ⟩ = ⟨ f_{2} (z, u), ψ ⟩ . \end{matrix}$ By using (6.2), we find that $\begin{matrix} ϵ_{n} | ⟨ θ^{n} (1, t), ψ ⟩ | ⩽ ϵ_{n} {‖ θ^{n} (1, t) ‖}_{2} ‖ ψ ‖_{2} ⩽ ϵ_{n} K ‖ ψ ‖_{2} \end{matrix}$ which implies that $\begin{matrix} lim_{n \to \infty} ϵ_{n} ⟨ θ^{n} (1, t), ψ ⟩ = 0 . \end{matrix}$ Therefore, we can pass to the limit in all terms in (6.6) to conclude that $\begin{matrix} ρ_{z} \frac{d}{d t} ⟨ z_{t}, φ ⟩ + ρ_{u} \frac{d}{d t} ⟨ u_{t}, ψ ⟩ + χ ⟨ u_{x}, ψ_{x} ⟩ + ⟨ \frac{a_{2}}{\sqrt{a_{1}}} u_{x} + \sqrt{a_{1}} z_{x}, \frac{a_{2}}{\sqrt{a_{1}}} ψ_{x} + \sqrt{a_{1}} φ_{x} ⟩ \\ + ⟨ z_{t}, φ ⟩ + ⟨ u_{t}, ψ ⟩ + ⟨ f_{1} (z, u) - h_{1}, φ ⟩ + ⟨ f_{2} (z, u) - h_{2}, ψ ⟩ = 0 . \end{matrix}$ Now, combining (6.7), (6.4) and (6.5) we deduce $\begin{matrix} θ (x, y, t) = u_{t} (x, t - τ y) a.e. in (0, L) \times (0, 1) \times (0, T) . \end{matrix}$ In addition, it is standard to show that $(z (0), u (0), z_{t} (0), u_{t} (0)) = (z_{0}, u_{0}, z_{1}, u_{1})$ . Therefore, the function $U = (z, u, z_{t}, u_{t}, θ)$ is a weak solution for the problem (2.3)–(2.4) with $ϵ = 0$ . The proof is complete. □

We are in position to formulate and prove the result on the upper semicontinuous convergence of global attractors.

Theorem 6.5.

Suppose the Assumption 2.1 holds, then the family of attractors $A_{ϵ}$ is upper semicontinuous with respect to $ϵ \to 0$ . More precisely, $\begin{matrix} (6.8) & lim_{ϵ \to 0} {dist}_{H} (A_{ϵ}, A_{0}) = 0 . \end{matrix}$

Proof.

We proceed by contradiction as in [19, 21, 23]. Suppose that (6.8) does not hold. Then, there exist an $δ > 0$ and a sequence $ϵ_{n} \to 0$ such that $\begin{matrix} {dist}_{H} (A_{ϵ_{n}}, A_{0}) ⩾ δ > 0, \forall n \in N . \end{matrix}$ Thus, there exists a sequence ${U_{0}^{n}} \in A_{ϵ_{n}}$ by the compactness of $A_{ϵ}$ such that $\begin{matrix} (6.9) & {dist}_{H} (U_{0}^{n}, A_{0}) ⩾ δ > 0, \forall n . \end{matrix}$ Let $U^{n} (t) = (z^{n} (t), u^{n} (t), z_{t}^{n} (t), u_{t}^{n} (t), θ^{n} (t))$ be a full trajectory from the attractor $A_{ϵ_{n}}$ such that $U^{n} (0) = U_{0}^{n}$ . By Lemma 6.2, we have $\begin{matrix} (6.10) & {‖ U^{n} (t) ‖}_{H} ⩽ R_{0} + 1 . \end{matrix}$ This together with the fact that $ϵ_{n} \in [0, 1)$ imply that coefficients in (5.1) do not depend on $ϵ_{n}$ . Then by Theorem 3.13 there exists $R_{1} > 0$ not depending on $ϵ_{n}$ such that $\begin{matrix} (6.11) & {‖ U_{t}^{n} (t) ‖}_{H} ⩽ R_{1} . \end{matrix}$ By using ${(2.3)}_{1}$ and ${(2.3)}_{2}$ with $ϵ = ϵ_{n}$ , we obtain $\begin{array}{l} (6.12) & {(a_{1} z^{n} + a_{2} u^{n})}_{x x} = ρ_{z} z_{t t}^{n} + z_{t} + f_{1} (z^{n}, u^{n}) - h_{1} \in L^{2} (0, L), \\ (6.13) & {(a_{3} u^{n} + a_{2} z^{n})}_{x x} = ρ_{u} u_{t t}^{n} + u_{t}^{n} + ϵ_{n} θ^{n} (1, t) + f_{2} (z^{n}, u^{n}) - h_{2} \in L^{2} (0, L) . \end{array}$ Hence, using (6.11), Lemma 6.3 and the fact that nonlinear terms are locally Lipschitz continuous, there exists $R^{'} > 0$ not depending on $ϵ_{n}$ such that $\begin{matrix} {‖ {(a_{1} z^{n} + a_{2} u^{n})}_{x x} ‖}_{2} + {‖ {(a_{3} u^{n} + a_{2} z^{n})}_{x x} ‖}_{2} ⩽ R^{'} . \end{matrix}$ Thus, since $\begin{matrix} z_{x x}^{n} = \frac{a_{3}}{a_{1} a_{3} - a_{2}^{2}} {(a_{1} z^{n} + a_{2} u^{n})}_{x x} - \frac{a_{2}}{a_{1} a_{3} - a_{2}^{2}} {(a_{3} u^{n} + a_{2} z^{n})}_{x x} \end{matrix}$ and $\begin{matrix} u_{x x}^{n} = - \frac{a_{2}}{a_{1} a_{3} - a_{2}^{2}} {(a_{1} z^{n} + a_{2} u^{n})}_{x x} + \frac{a_{1}}{a_{1} a_{3} - a_{2}^{2}} {(a_{3} u^{n} + a_{2} z^{n})}_{x x} \end{matrix}$ we obtain $R_{3} > 0$ not depending on $ϵ_{n}$ such that $\begin{matrix} (6.14) & {‖ z_{x x}^{n} ‖}_{2} + {‖ u_{x x}^{n} ‖}_{2} ⩽ R_{3} . \end{matrix}$ Combining (6.11) and (6.14) we conclude that $\begin{matrix} (6.15) & \begin{matrix} (z^{n} (t), u^{n} (t), z_{t}^{n} (t), u_{t}^{n} (t)) is uniformly bounded in L^{\infty} (R; V), \\ (z_{t}^{n} (t), u_{t}^{n} (t), z_{t t}^{n} (t), u_{t t}^{n} (t)) is uniformly bounded in L^{\infty} (R; H), \end{matrix} \end{matrix}$ where $\begin{matrix} V : = {(H^{2} (0, L) \cap H_{*}^{1} (0, L))}^{2} \times {(H_{*}^{1} (0, L))}^{2} and H : = {(H_{*}^{1} (0, L))}^{2} \times {(L^{2} (0, L))}^{2} . \end{matrix}$ Since $V$ is compactly embedded into $H$ , using Simon’s Compactness Theorem (see [35]), there exists a subsequence such that $\begin{matrix} (6.16) & (z^{n} (t), u^{n} (t), z_{t}^{n} (t), u_{t}^{n} (t)) \to (z (t), u (t), z_{t} (t), u_{t} (t)) in C ([- T, T]; H), \forall T > 0 . \end{matrix}$ We consider the variable θ given by $\begin{matrix} θ (y, t) = u_{t} (t - y τ), (y, t) \in (0, 1) \times R . \end{matrix}$ We will prove that $U (t) = (z (t), u (t), z_{t} (t), u_{t} (t), θ (t))$ is a full trajectory of the limiting semigroup $S_{0} (t)$ . Indeed, by (6.10) and (6.16), we conclude that $\begin{matrix} sup_{t \in R} {‖ U (t) ‖}_{H} ⩽ R_{0} + 1 . \end{matrix}$ Using the same argument as in the proof of Theorem 6.4 we conclude that U is a full trajectory for the limiting semi-flow $S_{0} (t)$ . Finally, the limit (6.16) implies $\begin{matrix} U_{0}^{n} \to U (0) \in A_{0}, \end{matrix}$ which is contradict (6.9). The proof is complete now. □

Footnotes

Acknowledgements

A.J.A. Ramos thanks the CNPq for financial support through Grant 310729/2019-0.

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