Limiting dynamics in one-dimensional porous-elasticity

Abstract

In this paper we investigate the long-term behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that the porous-elasticity problem converges to a quasi-static problem for the microvoids motion as a suitable parameter J tends to zero. Finite dimensional global attractor with additional regularity in J is obtained using the recent quasi-stability theory. Finally, we compare the porous-elasticity problem with quasi-static problem, in the sense of the upper-semicontinuity of their attractors as $J \to 0$ .

Keywords

Porous-elasticity quasi-static microvoids singular limit global attractor upper-semicontinuity

1. Introduction

For the last several decades, various types of equations have been employed as some mathematical model describing physical, chemical, biological and engineering systems. Among them, the mathematical models of vibrating, flexible structures has been considerably stimulated in recent years by an increasing number of questions of practical concern. Research on stabilization of distributed parameter systems has largely focused on the stabilization of dynamic models of individual structural members such as strings, membranes and beams (see [7] and references therein).

Elasticity problems also have attracted the attention of researchers from different fields interested in the temporal decay behavior of the solutions. This interest has given many results that can be seen in the literature. If elastic solids with voids are considered, as in this paper, one should look into the theory of porous-elastic materials. Here we deal with the theory established by Cowin and Nunziato (see [13]).

With this the mind, let us consider the evolution equations for one-dimensional theories of porous materials given by $\begin{matrix} (1.1) & \{\begin{matrix} ρ u_{t t} = T_{x}, \\ J ϕ_{t t} = H_{x} + G . \end{matrix} \end{matrix}$ Here T is the stress, H is the equilibrated stress and G is the equilibrated body force. The variables u and ϕ are, respectively, the displacement of the solid elastic material and the volume fraction. The constitutive equations are $\begin{matrix} (1.2) & \{\begin{matrix} T = μ u_{x} + b ϕ + γ u_{t x}, \\ H = δ ϕ_{x}, \\ G = - b u_{x} - ξ ϕ - τ ϕ_{t} . \end{matrix} \end{matrix}$ Here ρ, J, μ, b, γ, δ, ξ and τ, are the constitutive coefficients whose physical meaning is well known.

The constitutive coefficients, in one-dimensional case, satisfies $\begin{matrix} (1.3) & ξ > 0, δ > 0, μ > 0, ρ > 0, J > 0, τ > 0 and μ ξ > b^{2} . \end{matrix}$ As coupling is considered, b must be different from 0, but its sign does not matter in the analysis.

Substituting the constitutive equations (1.2) into the evolution equations (1.1), we arrive at $\begin{matrix} (1.4) & \{\begin{matrix} ρ u_{t t} - μ u_{x x} - b ϕ_{x} - γ u_{x x t} = 0 & in (0, L) \times (0, \infty), \\ J ϕ_{t t} - δ ϕ_{x x} + b u_{x} + ξ ϕ + τ ϕ_{t} = 0 & in (0, L) \times (0, \infty) . \end{matrix} \end{matrix}$ It is worth mentioning some papers in connection with the above system. In [9] A. Magaña and R. Quintanilla studied the system (1.4). They proved that the system (1.4) is exponentially stable using the semigroup arguments due to Liu and Zheng [8]. Also, they proved that when $τ = 0$ the obtained system is not exponentially stable. In [11] J. Muñoz Rivera and R. Quintanilla proved that when $τ = 0$ the energy is controlled by a rate decay of type $\frac{1}{t}$ . Moreover using a result on [14] they improved the polynomial rate of decay by taking more regular initial data. In [15], M. L. Santos et al. proved that the system (1.4) with $τ = 0$ has lack of exponential decay independent of any relation between the coefficients of wave propagation, and it decays as $\frac{1}{\sqrt{t}}$ . In addition they also proved that this rate is optimal.

Now, setting $J \to 0$ , the above system becomes into the system $\begin{matrix} (1.5) & \{\begin{matrix} ρ u_{t t} - μ u_{x x} - b ϕ_{x} - γ u_{x x t} = 0 & in (0, L) \times (0, \infty), \\ τ ϕ_{t} - δ ϕ_{x x} + b u_{x} + ξ ϕ = 0 & in (0, L) \times (0, \infty) . \end{matrix} \end{matrix}$ The system (1.5) was studied in [10]. There, using the energy method, the authors proved that the solutions are exponentially stable, as they were for the dynamical case. On the other hand, when the viscoelasticity is absent, but porous dissipation is present and the void motion is quasi-static, was proved in [12] that the solutions are polynomially stable.

In this present paper, we consider the nonlinear model $\begin{matrix} (1.6) & \{\begin{matrix} ρ u_{t t} - μ u_{x x} - b ϕ_{x} - γ u_{x x t} + f (u) = 0 & in (0, L) \times (0, \infty), \\ J ϕ_{t t} - δ ϕ_{x x} + b u_{x} + ξ ϕ + τ ϕ_{t} = 0 & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ where $f (u)$ is a nonlinear feedback force. The system is supplemented with the boundary conditions $\begin{matrix} (1.7) & u (0, t) = u (L, t) = ϕ_{x} (0, t) = ϕ_{x} (L, t) = 0, t ⩾ 0, \end{matrix}$ and initial conditions $\begin{matrix} (1.8) & \begin{matrix} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), \\ ϕ (x, 0) = ϕ_{0} (x), ϕ_{t} (x, 0) = ϕ_{1} (x), in (0, L) . \end{matrix} \end{matrix}$

Our objective in the present contribution is to establish the following results:

We rigorously prove that, as $J \to 0$ , solutions of the system of the porous-elasticity (1.6)–(1.8) converges to that of the quasi-static system $\begin{matrix} (1.9) & \{\begin{matrix} ρ u_{t t} - μ u_{x x} - b ϕ_{x} - γ u_{x x t} + f (u) = 0 & in (0, L) \times (0, \infty), \\ τ ϕ_{t} - δ ϕ_{x x} + b u_{x} + ξ ϕ = 0 & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ with boundary conditions $\begin{matrix} (1.10) & u (0, t) = u (L, t) = ϕ_{x} (0, t) = ϕ_{x} (L, t) = 0, \end{matrix}$ and initial condition $\begin{matrix} (1.11) & u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), ϕ (x, 0) = ϕ_{0} (x) . \end{matrix}$ The analysis of the above limit when $J \to 0$ is singular in the sense that at $J = 0$ the parabolic/hyperbolic system is obtained. See Theorem 3.1.

We prove the existence of a smooth finite dimensional attractor using the recente quasi-stability theory [1,3]. We get the uniform (in J) quasi-stability by establishing a observability inequality uniformly in J. See Theorem 4.12. We also prove the quasi-stability for the system (1.9) as limit of the uniform quasi-stability of the system (1.6) as $J \to 0$ . See Remark 4.13.

We establishes the relationship between the attractors of the system of the porous-elasticity with those of the quasi-static system. More precisely, we prove the upper-semicontinuity of attractors of (1.6)–(1.8) as $J \to 0$ . See Theorem 5.1.

The main difficulty above is to obtain uniform estimates with respect to J for full trajectories from the attractor of the system of the porous-elasticity. To derive the uniform smoothness we construct a ball absorbing independent of $0 < J < 1$ . See Remark 4.10.

2. Functional setting and well-posedness

We introduce the phase space to study the well-posedness and dynamic problem (1.6). We denote the inner product and norm in $L^{2} (0, L)$ by $⟨ \cdot, \cdot ⟩$ and $‖ \cdot ‖$ , respectively. Let us consider the spaces $\begin{matrix} L_{*}^{2} (0, L) : = {u \in L^{2} (0, L); \int_{0}^{L} u (x) d x = 0}, \end{matrix}$ and $\begin{matrix} (2.1) & H_{*}^{1} (0, L) : = H^{1} (0, L) \cap L_{*}^{2} (0, L), H_{*}^{2} (0, L) : = H^{2} (0, L) \cap H_{*}^{1} (0, L) . \end{matrix}$ We define the phase space $\begin{matrix} (2.2) & H = H_{0}^{1} (0, L) \times H_{*}^{1} (0, L) \times L^{2} (0, L) \times L_{*}^{2} (0, L), \end{matrix}$ endowed with the J-dependent norm $\begin{matrix} {‖ (u, ϕ, v, φ) ‖}_{H_{J}}^{2} = ρ ‖ v ‖^{2} + J ‖ φ ‖^{2} + μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩ . \end{matrix}$ Taking into account that $μ ξ > b^{2}$ , there exists a constant $ϱ_{0} > 0$ such that $\begin{matrix} (2.3) & ‖ u_{x} ‖^{2} + ‖ ϕ_{x} ‖^{2} ⩽ ϱ_{0} (μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩) . \end{matrix}$ Using the Poincaré’s inequality and (2.3), there exists a constant $ϱ > 0$ such that $\begin{matrix} (2.4) & ‖ u ‖^{2} + ‖ ϕ ‖^{2} ⩽ ϱ (μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩) . \end{matrix}$

Concerning the nonlinear source terms we assume the following

There exist $C_{f} > 0$ and $p ⩾ 1$ such that $\begin{matrix} (2.5) & | f^{'} (u) | ⩽ C_{f} (1 + | u |^{p - 1}), \forall u \in R . \end{matrix}$

Denoting $F (u) = \int_{0}^{u} f (s) d s$ , there exists a constant $C_{F} > 0$ such that $\begin{matrix} (2.6) & F (u) + C_{F} ⩾ 0 and f (u) u - F (u) + C_{F} ⩾ 0, \forall u \in R . \end{matrix}$

We denote $y (t) = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t))$ and $y_{0} = (u_{0}, ϕ_{0}, u_{1}, ϕ_{1})$ , then the problem (1.6) can be rewritten as a Cauchy problem $\begin{matrix} (2.7) & \{\begin{matrix} \frac{d y (t)}{d t} + L y (t) = F (y (t)), t > 0, \\ y (0) = y_{0} \in H, \end{matrix} \end{matrix}$ where $L : D (L) \subset H \to H$ is defined by $\begin{matrix} L (\begin{matrix} u \\ ϕ \\ v \\ φ \end{matrix}) = (\begin{matrix} - v \\ - φ \\ \frac{1}{ρ} (- μ u_{x x} - b ϕ_{x} - γ v_{x x}) \\ \frac{1}{J} (- δ ϕ_{x x} + b u_{x} + ξ ϕ + τ φ) \end{matrix}), \end{matrix}$ with domain $\begin{matrix} D (L) = {y \in H : ϕ \in H^{2} (0, L), v \in H_{0}^{1} (0, L), φ \in H^{1} (0, L), μ u + γ v \in H^{2} (0, L)} . \end{matrix}$ The function $F : H \to H$ is defined by $\begin{matrix} (2.8) & F (\begin{matrix} u \\ ϕ \\ v \\ φ \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ - \frac{1}{ρ} f (u) \\ 0 \end{matrix}) . \end{matrix}$

We define, along a strong solution, the total energy by $\begin{matrix} (2.9) & E_{J} (t) = E_{J} (t) + \int_{0}^{L} F (u (t)) d t, \end{matrix}$ where $E_{J} (t)$ is defined by $\begin{matrix} E_{J} (t) = \frac{1}{2} {‖ y (t) ‖}_{H_{J}}^{2} = \frac{1}{2} {‖ (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t)) ‖}_{H_{J}}^{2} . \end{matrix}$

Now, we need the following auxiliary result to be used in the sequel.

Lemma 2.1.
Assume that assumptions (A1) and (A2) hold and that $y (t) = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t))$ is a strong solution of ( 1.6 ), then $\begin{matrix} (2.10) & \frac{d}{d t} E_{J} (t) = - γ {‖ u_{x t} (t) ‖}^{2} - τ {‖ ϕ_{t} (t) ‖}^{2}, \forall t ⩾ 0 . \end{matrix}$ Moreover, $\begin{matrix} (2.11) & \frac{1}{2} {‖ y (t) ‖}_{H_{J}}^{2} - C_{F} L ⩽ E_{J} (t) ⩽ C_{F} (1 + {‖ y (t) ‖}_{H_{J}}^{p + 1}), \forall t ⩾ 0 . \end{matrix}$
Proof.
Applying the product in $L^{2} (0, L)$ to the equations in (1.6) with $u_{t}$ and $ϕ_{t}$ , respectively, we can deduce that (2.10) holds. It follows from (2.6) that $\begin{matrix} E_{J} (t) ⩾ \frac{1}{2} {‖ y (t) ‖}_{H_{J}}^{2} - C_{F} L . \end{matrix}$ Hence, the first inequality in (2.11) holds. On the other hand, from assumption (2.5) and the embedding $H_{0}^{1} (0, L) ↪ L^{\infty} (0, L)$ , we infer that $\begin{matrix} \int_{0}^{L} F (u) d x ⩽ C_{F} (1 + ‖ u_{x} ‖^{p + 1}), \end{matrix}$ which implies the second inequality in (2.11). The proof is complete. □
Lemma 2.2.
The operator $L$ is maximal monotone.
Proof.
Let $y = (u, ϕ, v, φ)$ , $\tilde{y} = (\tilde{u}, \tilde{ϕ}, \tilde{v}, \tilde{φ}) \in D (L)$ . Then, it is straightforward see that $\begin{matrix} {⟨ L y - L \tilde{y}, y - \tilde{y} ⟩}_{H_{J}} = γ ‖ v_{x} - {\tilde{v}}_{x} ‖^{2} + τ ‖ φ - \tilde{φ} ‖^{2} . \end{matrix}$ Therefore $L$ is monotone. In order to prove that $L$ is maximal monotone, we need to prove that $Range (I + L) = H$ . For this, it is sufficient to show that for $h = (h_{1}, h_{2}, h_{3}, h_{4}) \in H$ , there exists $y = (u, ϕ, v, φ) \in D (L)$ such that $y + L y = h$ , that is, $\begin{matrix} (2.12) & \{\begin{matrix} u - v = h_{1}, \\ ϕ - φ = h_{2}, \\ ρ v - μ u_{x x} - b ϕ_{x} - γ v_{x x} = ρ h_{3}, \\ J φ - δ ϕ_{x x} + b u_{x} + ξ ϕ + τ φ = J h_{4}, \end{matrix} \end{matrix}$ which is equivalent to $\begin{matrix} (2.13) & \{\begin{matrix} ρ u - (μ + γ) u_{x x} - b ϕ_{x} = ρ (h_{1} + h_{3}) - γ h_{1 x x}, \\ (J + ξ + τ) ϕ - δ ϕ_{x x} + b u_{x} = J (h_{2} + h_{4}) + τ h_{2}, \end{matrix} \end{matrix}$ and the system (2.13) is equivalent to the variational problem $\begin{matrix} B ((u, ϕ), (\tilde{u}, \tilde{ϕ})) = C (\tilde{u}, \tilde{ϕ}), \end{matrix}$ where the bilinear form $B : {(H_{0}^{1} (0, L))}^{2} \times {(H_{}^{1} (0, L))}^{2} \to R$ and the linear form $C : H_{0}^{1} (0, L) \times H_{}^{1} (0, L) \to R$ are given by $\begin{matrix} (2.14) & \begin{matrix} \begin{matrix} B ((u, ϕ), (\tilde{u}, \tilde{ϕ})) & = ρ ⟨ u, \tilde{u} ⟩ + (J + ξ + τ) ⟨ ϕ, \tilde{ϕ} ⟩ + (μ + γ) ⟨ u_{x}, {\tilde{u}}_{x} ⟩ + δ ⟨ ϕ_{x}, {\tilde{ϕ}}_{x} ⟩ \\ + b (⟨ ϕ, {\tilde{u}}_{x} ⟩ + ⟨ u_{x}, \tilde{ϕ} ⟩), \end{matrix} \\ C (\tilde{φ}, \tilde{ψ}) = ρ ⟨ h_{1} + h_{3}, \tilde{u} ⟩ + J ⟨ h_{2} + h_{4}, \tilde{ϕ} ⟩ + γ ⟨ h_{1 x}, {\tilde{u}}_{x} ⟩ + τ ⟨ h_{2}, \tilde{ϕ} ⟩ . \end{matrix} \end{matrix}$ It is easy see that $B$ and $C$ are continuous, and moreover, $B$ is coercive since (2.3) implies $\begin{matrix} B ((u, ϕ), (u, ϕ)) & = ρ ‖ u ‖^{2} + (J + ξ + τ) ‖ ϕ ‖^{2} + (μ + γ) ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩ \\ ⩾ μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩ \\ ⩾ ϱ_{0}^{- 1} (‖ u_{x} ‖^{2} + ‖ ϕ_{x} ‖^{2}) . \end{matrix}$ Thus, by Lax–Milgram Theorem, the system (2.13) has a unique weak solution $(u, ϕ) \in H_{0}^{1} (0, L) \times H_{}^{1} (0, L)$ which along with (2.12) implies $\begin{matrix} (2.15) & \{\begin{matrix} v = u - h_{1} \in H_{0}^{1} (0, L), \\ φ = ϕ - h_{2} \in H_{}^{1} (0, L), \\ {(μ u + γ v)}_{x x} = ρ v - b ϕ_{x} - ρ h_{3} \in L^{2} (0, L), \\ δ ϕ_{x x} = J φ + b u_{x} + ξ ϕ + τ φ - J h_{4} \in L^{2} (0, L), \end{matrix} \end{matrix}$ which implies that $y = (φ, ϕ, u, v) \in D (L)$ and satisfies $y + L y = h$ . This complete the proof of the maximal monotonicity of $L$ . □
Theorem 2.3.
Suppose that the assumptions (A1) and (A2) hold. Then
Given $y_{0} = (u_{0}, ϕ_{0}, u_{1}, ϕ_{1}) \in H$ , then problem ( 2.7 ) has a unique global weak solution satisfying $y (t) \in C ([0, \infty); H)$ . Moreover, this solution satisfies the estimate $\begin{matrix} (2.16) & sup_{t ⩾ 0} {‖ y (t) ‖}_{H_{J}}^{2} + \int_{0}^{\infty} ({‖ u_{x t} (s) ‖}^{2} + {‖ ϕ_{t} (s) ‖}^{2}) d s ⩽ C_{F} (1 + {‖ y (0) ‖}_{H_{J}}^{p + 1}) . \end{matrix}$

If $y_{0} = (u_{0}, ϕ_{0}, u_{1}, ϕ_{1}) \in D (L)$ , then the corresponding solution is strong. Moreover, the weak solutions depend continuously on the initial data $y_{0}$ in $H$ .

Proof.
We need to prove first that $F$ defined in (2.8) is locally Lipschitz in $H$ . Let us denote $y^{1} = (u^{1}, ϕ^{1}, v^{1}, φ^{1}) \in H$ and $y^{2} = (u^{2}, ϕ^{2}, v^{2}, φ^{2}) \in H$ such that $‖ y^{1} ‖_{H_{J}}$ , $‖ y^{2} ‖_{H_{J}} ⩽ R$ , where $R > 0$ . We have from the mean value theorem and assumption (2.5) that $\begin{matrix} {| f (u^{1}) - f (u^{2}) |}^{2} = {| f^{'} (ϱ u^{1} + (1 - ϱ) u^{2}) |}^{2} \cdot {| u^{1} - u^{2} |}^{2} ⩽ C_{f}^{2} {(1 + {| u^{1} |}^{p - 1} + {| u^{2} |}^{p - 1})}^{2} {| u^{1} - u^{2} |}^{2}, \end{matrix}$ where $0 ⩽ ϱ ⩽ 1$ . Hence, from the embedding $H_{0}^{1} (0, L) ↪ L^{\infty} (0, L)$ , we infer that, for some $C_{R} > 0$ , $\begin{matrix} {‖ F (y^{1}) - F (y^{1}) ‖}_{H_{J}}^{2} ⩽ C \int_{0}^{L} {| f (u^{1}) - f (u^{2}) |}^{2} d x ⩽ C_{R} {‖ u^{1} - u^{2} ‖}^{2} ⩽ C_{R} {‖ y^{1} - y^{2} ‖}_{H_{J}}^{2} . \end{matrix}$ Therefore, $F$ is locally Lipschitz continuous on $H$ .

Since $L$ is maximal monotone and $F$ is locally Lipschitz continuous, then by [4, Theorem 7.2], for all $y_{0} \in D (L)$ , problem (2.7) has a unique strong solution y defined on a maximal interval $[0, t_{max})$ with $t_{max} ⩽ \infty$ . If $y_{0} \in H$ , then (2.7) has a unique weak solution $y \in C ([0, t_{max}); H)$ . In addition, if $t_{max} < \infty$ , then ${lim sup}_{t \to t_{max}^{-}} ‖ y (t) ‖_{H} = \infty$ .

Next we prove the global existence of strong solutions, that is, $t_{max} = \infty$ . From (2.10) and (2.11), we find that $\begin{array}{l} \frac{1}{2} {‖ y (t) ‖}_{H_{J}}^{2} + \int_{0}^{t} ({‖ u_{x t} (s) ‖}^{2} + {‖ ϕ_{t} (s) ‖}^{2}) d s & ⩽ E_{J} (t) + \int_{0}^{t} ({‖ u_{x t} (s) ‖}^{2} + {‖ ϕ_{t} (s) ‖}^{2}) d s + C_{F} L \\ = E_{J} (0) + C_{F} L \\ (2.17) & ⩽ C_{F} (1 + {‖ y (0) ‖}_{H_{J}}^{p + 1}) + C_{F} L, \forall t ⩾ 0, \end{array}$ which implies the global existence of strong solutions. By using density arguments, we can see that (2.17) also holds for weak solutions. This shows the global existence of weak solutions. Moreover, estimate (2.16) holds for these solutions.

Moreover, it is not difficult to check that for any $y^{1} (0)$ , $y^{2} (0) \in H$ , the corresponding global solutions $y^{1} (t)$ , $y^{2} (t)$ satisfy $\begin{matrix} (2.18) & {‖ y^{1} (t) - y^{2} (t) ‖}_{H}^{2} ⩽ C_{T} {‖ y^{1} (0) - y^{2} (0) ‖}_{H}^{2}, 0 ⩽ t ⩽ T, \end{matrix}$ for all $T > 0$ , where $C_{T}$ is a constant depending on the norms of $y^{1} (0), y^{2} (0) \in H$ and T. Moreover, (2.18) implies the continuous dependence of the weak solution on the initial data. The proof is complete. □
Remark 2.4.
To study the quasi-static problem (1.9)–(1.11), we consider the Hilbert space $\begin{matrix} (2.19) & H_{0} = H_{0}^{1} (0, L) \times L^{2} (0, L) \times H_{*}^{1} (0, L), \end{matrix}$ endowed with the norm $\begin{matrix} ‖ z ‖_{H_{0}}^{2} = ρ ‖ v ‖^{2} + μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩, z = (u, v, ϕ) \in H_{0} . \end{matrix}$ In this case, the same argument used in the proof of Theorem 2.3 shows that the quasi-static problem (1.9)–(1.11) is well-posed in the phase space $H_{0}$ .

Now, we define the total energy of solutions $z (t) = (u (t), u_{t} (t), ϕ (t))$ of the quasi-static problem (1.9)–(1.11) by $\begin{matrix} (2.20) & E (t) = E (t) + \int_{0}^{L} F (u (t)) d t, \end{matrix}$ where $E (t)$ is the energy defined by $\begin{matrix} (2.21) & E (t) = \frac{1}{2} {‖ z (t) ‖}_{H_{0}}^{2} = {‖ (u (t), u_{t} (t), ϕ (t)) ‖}_{H_{0}}^{2} . \end{matrix}$ Remark 2.5.
By Theorem 2.3 the problem of the porous-elasticity (1.6)–(1.8) generates the dynamical system $(H, S_{J} (t))$ with the phase space $H$ given in (2.2) and the associated $C_{0}$ semigroup $S_{J} (t)$ given by $\begin{matrix} (2.22) & S_{J} (t) y_{0} = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t)), y_{0} = (u_{0}, ϕ_{0}, u_{1}, ϕ_{1}) \in H, \end{matrix}$ where $(u (t), ϕ (t), u_{t} (t), ϕ_{t} (t))$ is a weak solution to (1.6)–(1.8). Similarly, it follows from Remark 2.4 that we can define the dynamical system $(H_{0}, S (t))$ with $H_{0}$ given in (2.19) and $S (t)$ given by $\begin{matrix} (2.23) & S (t) z_{0} = (u (t), u_{t} (t), ϕ (t)), z_{0} = (u_{0}, u_{1}, ϕ_{0}) \in H_{0}, \end{matrix}$ where $(u (t), u_{t} (t), ϕ (t))$ solves the quasi-static problem (1.9)–(1.11).

3. Asymptotic limit as $J \to 0$

In this section, we study the asymptotic limit of the system of the porous-elasticity (1.6)–(1.8) as $J \to 0$ . The main result of this section is stated as follows.

Theorem 3.1.
Suppose that assumptions (A1) and (A2) hold. Let $y^{J} = (u^{J}, ϕ^{J}, u_{t}^{J}, ϕ_{t}^{J})$ be a weak solution to ( 1.6 ) with initial condition $y_{0} = (u_{0}, ϕ_{0}, u_{1}, ϕ_{1}) \in H$ . Then for any $T > 0$ , as $J \to 0$ , we have $\begin{matrix} (u^{J}, ϕ^{J}, u_{t}^{J}, ϕ_{t}^{J}) \to (u, ϕ, u_{t}, ϕ_{t}) weakly star in L^{\infty} (0, T; H), \end{matrix}$ where $(u, ϕ)$ is a weak solution to quasi-static problem ( 1.9 ).
Proof.
Using the estimate (2.16), we deduce $\begin{matrix} (3.1) & sup_{t ⩾ 0} {‖ y^{J} (t) ‖}_{H_{J}}^{2} + \int_{0}^{\infty} ({‖ u_{x t}^{J} (s) ‖}^{2} + {‖ ϕ_{t}^{J} (s) ‖}^{2}) d s ⩽ C, \end{matrix}$ where the constant $C > 0$ independe of $0 < J < 1$ . Then, using (2.3) and (3.1), we obtain that $\begin{matrix} (u_{t}^{J}), (\sqrt{J} ϕ_{t}^{J}), (u_{x}^{J}), (ϕ_{x}^{J}) are bounded in L^{\infty} (0, T; L^{2} (0, L)), \\ (u_{x t}^{J}), (ϕ_{t}^{J}) are bounded in L^{2} (0, T; L^{2} (0, L)) . \end{matrix}$ Extracting subsequences, without changing notation, one gets $\begin{array}{l} (3.2) & u^{J} \to u weakly star in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ (3.3) & ϕ^{J} \to ϕ weakly star in L^{\infty} (0, T; H_{}^{1} (0, L)), \\ (3.4) & u_{t}^{J} \to u_{t} weakly star in L^{\infty} (0, T; L^{2} (0, L)) \cap L^{2} (0, T; H_{0}^{1} (0, L)), \\ (3.5) & ϕ_{t}^{J} \to ϕ_{t} weakly star in L^{2} (0, T; L^{2} (0, L)), \\ (3.6) & \sqrt{J} ϕ_{t}^{J} \to ϑ weakly star in L^{\infty} (0, T; L^{2} (0, L)) . \end{array}$ Now, using a compactness theorem due to Aubin and Lions (see, e.g. [16], Corollary 4), we obtain $\begin{matrix} (3.7) & \begin{matrix} u^{J} \to u strongly in C (0, T; H_{0}^{1 - ϵ} (0, L)), \\ ϕ^{J} \to ϕ strongly in C (0, T; H^{1 - ϵ} (0, L)), \end{matrix} \end{matrix}$ for all $ϵ > 0$ small enough. Combining (3.7) with (2.5), it follows that $\begin{matrix} (3.8) & f (u^{J}) \to f (u) strongly in L^{2} (0, T; L^{2} (0, L)) . \end{matrix}$ Moreover, using (3.6) we get $\begin{matrix} (3.9) & J ϕ_{t}^{J} = \sqrt{J} \sqrt{J} ϕ_{t}^{J} \to 0 weakly star in L^{\infty} (0, T; L^{2} (0, L)) . \end{matrix}$ Then, it remains to show that $(u, ϕ)$ is a weak solution to quasi-static problem (1.9)–(1.11). Indeed, first note that the variational formulation of problem (1.6)–(1.8) is given by $\begin{matrix} (3.10) & \begin{matrix} ρ \frac{d}{d t} ⟨ u_{t}^{J}, v ⟩ + J \frac{d}{d t} ⟨ ϕ_{t}^{J}, w ⟩ + μ ⟨ u_{x}^{J}, v_{x} ⟩ + δ ⟨ ϕ_{x}^{J}, w_{x} ⟩ + ξ ⟨ ϕ^{J}, w ⟩ + b (⟨ ϕ^{J}, v_{x} ⟩ + ⟨ u_{x}^{J}, w ⟩) \\ + γ ⟨ u_{x t}^{J}, v_{x} ⟩ + τ ⟨ ϕ_{t}^{J}, w ⟩ + ⟨ f (u^{J}), v ⟩ = 0 . \end{matrix} \end{matrix}$ for all $v \in H_{0}^{1} (0, L)$ and $w \in H_{}^{1} (0, L)$ . Using convergences (3.2)–(3.5), (3.8) and (3.9) in equation (3.10), one obtains the weak formulation of the system (1.9)–(1.11) given by $\begin{matrix} (3.11) & \begin{matrix} ρ \frac{d}{d t} ⟨ u_{t}, v ⟩ + τ ⟨ ϕ_{t}, w ⟩ + μ ⟨ u_{x}, v_{x} ⟩ + δ ⟨ ϕ_{x}, w_{x} ⟩ + ξ ⟨ ϕ, w ⟩ + b (⟨ ϕ, v_{x} ⟩ + ⟨ u_{x}, w ⟩) \\ + γ ⟨ u_{x t}, v_{x} ⟩ + ⟨ f (u), v ⟩ = 0 . \end{matrix} \end{matrix}$ for all $u \in H_{0}^{1} (0, L)$ and $w \in H_{*}^{1} (0, L)$ . Thus to conclude the proof of we only need to show that the $(u (0), u_{t} (0), ϕ (0)) = (u_{0}, u_{1}, ϕ_{0})$ . It follows from (3.7) that $u (0) = u_{0}$ and $ϕ (0) = ϕ_{0}$ . In order to identify $u_{t} (0) = u_{1}$ , multiply both sides of (3.10) by the teste function $θ \in H^{1} (0, T)$ with $θ (0) = 1$ and $θ (T) = 0$ , and integrate by parts over $[0, T]$ to obtain $\begin{matrix} - ρ ⟨ u_{1}, v ⟩ - ρ \int_{0}^{T} ⟨ u_{t}^{J}, v ⟩ θ_{t} d t - J ⟨ ϕ_{1}, w ⟩ - J \int_{0}^{T} ⟨ ϕ_{t}^{J}, w ⟩ θ_{t} d t + μ \int_{0}^{T} ⟨ u_{x}^{J}, v_{x} ⟩ θ d t \\ + δ \int_{0}^{T} ⟨ ϕ_{x}^{J}, w_{x} ⟩ θ d t + ξ \int_{0}^{T} ⟨ ϕ^{J}, w ⟩ θ d t + b \int_{0}^{T} (⟨ ϕ^{J}, v_{x} ⟩ + ⟨ u_{x}^{J}, w ⟩) θ d t \\ + γ \int_{0}^{T} ⟨ u_{x t}^{J}, v_{x} ⟩ θ d t + τ \int_{0}^{T} ⟨ ϕ_{t}^{J}, w ⟩ θ d t \\ + \int_{0}^{T} ⟨ f (u^{J}), v ⟩ θ d t = 0 . \end{matrix}$ Taking the limit as $J \to 0$ , we obtain $\begin{matrix} (3.12) & \begin{matrix} - ρ ⟨ u_{1}, v ⟩ - ρ \int_{0}^{T} ⟨ u_{t}, v ⟩ θ_{t} d t + μ \int_{0}^{T} ⟨ u_{x}, v_{x} ⟩ θ d t \\ + δ \int_{0}^{T} ⟨ ϕ_{x}, w_{x} ⟩ θ d t + ξ \int_{0}^{T} ⟨ ϕ, w ⟩ θ d t + b \int_{0}^{T} (⟨ ϕ, v_{x} ⟩ + ⟨ u_{x}, w ⟩) θ d t \\ + γ \int_{0}^{T} ⟨ u_{x t}, v_{x} ⟩ θ d t + τ \int_{0}^{T} ⟨ ϕ_{t}, w ⟩ θ d t \\ + \int_{0}^{T} ⟨ f (u), v ⟩ θ d t = 0 . \end{matrix} \end{matrix}$ On the other hand, multiplying (3.11) by θ and integrating the result over $[0, T]$ , we obtain $\begin{array}{l} - ρ ⟨ u_{t} (0), v ⟩ - ρ \int_{0}^{T} ⟨ u_{t}, v ⟩ θ_{t} d t + μ \int_{0}^{T} ⟨ u_{x}, v_{x} ⟩ θ d t \\ + δ \int_{0}^{T} ⟨ ϕ_{x}, w_{x} ⟩ θ d t + ξ \int_{0}^{T} ⟨ ϕ, w ⟩ θ d t + b \int_{0}^{T} (⟨ ϕ, v_{x} ⟩ + ⟨ u_{x}, w ⟩) θ d t \\ + γ \int_{0}^{T} ⟨ u_{x t}, v_{x} ⟩ θ d t + τ \int_{0}^{T} ⟨ ϕ_{t}, w ⟩ θ d t \\ (3.13) & + \int_{0}^{T} ⟨ f (u), v ⟩ θ d t = 0 . \end{array}$ Combining (3.12) and (3.13) we conclude that $u_{t} (0) = u_{1}$ . The proof is complete. □

4. Global attractors

4.1. Attractors for quasi-stable dissipative systems. Abstract results

In this subsection, we present some definitions related to global attractors for quasi-stable systems that can be found in recent references such as [1,3]. A dynamical system is a pair $(H, S (t))$ , where H is a Banach space and $S (t)$ is a continuous semigroup defined on H. We recall that a set $A \subset H$ is a global attractor for $(H, S (t))$ if it is compact, invariant, that is, $S (t) A = A$ for all $t ⩾ 0$ , and uniformly attracting, that is, $\begin{matrix} lim_{t \to \infty} {dist}_{H} (S (t) B, A) = 0, \end{matrix}$ for any bounded set $B \subset H$ , where ${dist}_{H}$ is the Hausdorff semi-distance in H.

As is well known in the literature, the existence of a global attractor is granted under suitable dissipativeness and compactness conditions. A dynamical system is called dissipative if it admits a bounded absorbing set, that is, a bounded set $B \subset H$ such that, for any bounded set $B \subset H$ , there exists a time $T_{B} > 0$ satisfying $\begin{matrix} S (t) B \subset B, \forall t ⩾ T_{B} . \end{matrix}$ A dynamical system is called asymptotically smooth if, for any bounded set $B \subset H$ forward-invariant $(S (t) B \subset B, t ⩾ 0)$ , there exists a compact set $K \subset \overline{B}$ that uniformly attracts B. Then we have the following classical result (see [1,3]).

Theorem 4.1.
Let $(H, S (t))$ be a dynamical system dissipative and asymptotically smooth. Then it possesses a unique compact global attractor.

To obtain the asymptotically smooth property, we shall introduce the concept of quasi-stability [3, Chapter 7, Definition 7.9.2].
Definition 4.2.
Let X, Y and Z be three reflexive Banach spaces with X compactly embedded in Y. We consider the space $H = X \times Y \times Z$ with the norm $\begin{matrix} ‖ y ‖_{H}^{2} = ‖ ξ_{0} ‖_{X}^{2} + ‖ ξ_{1} ‖_{Y}^{2} + ‖ ϖ_{0} ‖_{Z}^{2}, y = (ξ_{0}, ξ_{1}, ϖ_{0}) \in H . \end{matrix}$ The trivial case $Z = {0}$ is allowed. We assume that the dynamical system $(H, S (t))$ is given by $\begin{matrix} (4.1) & S (t) y = (ξ (t), ξ_{t} (t), ϖ (t)), y = (ξ (0), ξ_{t} (0), ϖ (0)) \in H, \end{matrix}$ where the functions ξ and φ possess the properties $\begin{matrix} (4.2) & ξ \in C (R^{+}; X) \cap C^{1} (R^{+}; Y), ϖ \in C (R^{+}; Z) . \end{matrix}$ The dynamical system $(H, S (t))$ is called quasi-stable on a set $B \subset H$ if there exists a compact seminorm $μ_{X}$ on the space X and nonnegative scalar functions a and c, locally bounded in $[0, \infty)$ , and $b \in L^{1} (0, \infty)$ , with ${lim}_{t \to \infty} b (t) = 0$ , such that $\begin{matrix} (4.3) & {‖ S (t) y_{1} - S (t) y_{2} ‖}_{H}^{2} ⩽ a (t) ‖ y_{1} - y_{2} ‖_{H}^{2}, \end{matrix}$ and $\begin{matrix} (4.4) & {‖ S (t) y_{1} - S (t) y_{2} ‖}_{H}^{2} ⩽ b (t) ‖ y_{1} - y_{2} ‖_{H}^{2} + c (t) sup_{0 ⩽ s ⩽ t} {[μ_{X} (ξ^{1} (s) - ξ^{2} (s))]}^{2}, \end{matrix}$ for any $y_{1}, y_{2} \in B$ . Here we denote $S (t) y_{i} = (ξ^{i} (t), ξ_{t}^{i} (t), ϖ^{i} (t))$ , $i = 1, 2$ .

The following result, which can be found in [3, Proposition 7.9.4], show that the quasi-stability implies the asymptotic smoothness of the dynamical system.
Theorem 4.3.
Let $(H, S (t))$ be a dynamical system given by ( 4.1 ) and satisfying ( 4.2 ). Then $(H, S (t))$ is asymptotically smooth if it is quasi-stable on every bounded positively invariant set of H.
Definition 4.4.
The fractal dimension of a compact set M in H is defined by $\begin{matrix} (4.5) & {dim}_{f}^{H} M : = lim_{ϵ \to 0} sup \frac{ln n (M, ϵ)}{ln (1 / ϵ)}, \end{matrix}$ where $n (M, ϵ)$ is the minimal number of closed balls of radius ϵ which cover M.

The quasi-stability also implies the smoothness and finite dimensionality of the attractor (see [3, Theorems 7.9.6 and 7.9.8]).
Theorem 4.5.
Let $(H, S (t))$ be a dynamical system given by ( 4.1 ) and satisfying ( 4.2 ). Suppose that it has a global attractor $A$ . If $(H, S (t))$ is quasi-stable on $A$ , then we have:
The global attractor $A$ has finite fractal dimension $({dim}_{f}^{H} A < \infty)$ .

If the coefficient $c (t)$ in ( 4.4 ) is uniformly bounded, then any full trajectory $y (t) = (ξ (t), ξ_{t} (t), ϖ (t))$ in $A$ has the following regularity properties (in time) $\begin{matrix} u_{t} \in L^{\infty} (R; X) \cap C (R; Y), u_{t t} \in L^{\infty} (R; Y), ϖ_{t} \in L^{\infty} (R; Z) . \end{matrix}$ Moreover, there exists $R > 0$ such that $\begin{matrix} {‖ u_{t} (t) ‖}_{X}^{2} + {‖ u_{t t} (t) ‖}_{Y}^{2} + {‖ ϖ_{t} (t) ‖}_{Z}^{2} ⩽ R^{2}, \forall t \in R, \end{matrix}$ where R depends on the constant ${sup}_{t ⩾ 0} c (t) < \infty$ .

The next theorem concerns global attractors for the dynamical system $(H, S_{J} (t))$ given in (2.22).
Theorem 4.6.
Assume that assumptions ( 2.5 ) and ( 2.6 ) are satisfied. Then,
Global attractors: The dynamical system $(H, S_{J} (t))$ possesses a global attractor $A_{J}$ .

Finite-dimensionality: The global attractor $A_{J}$ has finite fractal dimension ${dim}_{f}^{H} A_{J}$ with upper bound independent of $0 < J < 1$ .

Regularity: The global attractor $A_{J}$ is bounded in $\begin{matrix} V = H_{0}^{1} (0, L) \times H_{}^{2} (0, L) \times H_{0}^{1} (0, L) \times H_{}^{1} (0, L), \end{matrix}$ with a bound independent of $0 < J < 1$ . Moreover, every full trajectory $y (t) = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t))$ from the attractor satisfies $\begin{array}{l} sup_{t \in R} ({‖ u_{t} (t) ‖}^{2} + J {‖ ϕ_{t} (t) ‖}^{2} + {‖ u_{x} (t) ‖}^{2} + {‖ ϕ_{x} (t) ‖}^{2} + \int_{- \infty}^{\infty} ({‖ u_{x t} (s) ‖}^{2} + {‖ ϕ_{t} (s) ‖}^{2}) d s) \\ (4.6) & ⩽ R, \end{array}$ and $\begin{array}{l} sup_{t \in R} ({‖ {(μ u (t) + γ u_{t} (t))}_{x x} ‖}^{2} + {‖ ϕ_{x x} (t) ‖}^{2} + {‖ u_{t t} (t) ‖}^{2} + J {‖ ϕ_{t t} (t) ‖}^{2} + {‖ u_{x t} (t) ‖}^{2} + {‖ ϕ_{x t} (t) ‖}^{2}) \\ (4.7) & ⩽ R, \end{array}$ where the constant $R > 0$ is independent of $0 < J < 1$ .

The proof of Theorem 4.6 is based on the recente quasi-stability theory by Chueschov and Lasiecka [1,3]. We first prove that the corresponding system is quasi-stable on a absorbing ball in the sense of Definition 4.2, and then we apply the Theorems 4.1 and 4.5. Remark 4.7.
We emphasize that the dissipativity and quasi-stability estimates of our problem are obtained uniformly with respect to parameter $0 < J < 1$ .

4.2. Uniform dissipativity as $J \to 0$

In this section, we construct a absorbing ball uniformly with respect to $0 < J < 1$ by using the standard Lyapunov’s method.

Theorem 4.8.
Suppose that hypotheses of Theorem 4.6 hold, then there exists a positive constant ω independent of $0 < J < 1$ such that the energy function $E_{J} (t)$ defined by ( 2.9 ) verifies $\begin{matrix} (4.8) & E_{J} (t) ⩽ (3 E_{J} (0) + C_{F} L) e^{- \frac{2 ω}{3} t} + 5 C_{F} L, \forall t ⩾ 0 . \end{matrix}$
Remark 4.9.
As a limit of (4.8) when $J \to 0$ , we obtain the stability of the total energy holds for $E (t)$ associated to the quasi-static problem (1.9) defined by (2.20), that is, $\begin{matrix} (4.9) & E (t) ⩽ (3 E (0) + C_{F} L) e^{- \frac{2 ω}{3} t} + 5 C_{F} L, \forall t ⩾ 0 . \end{matrix}$ Note also that, for $C_{F} = 0$ , the energy $E (t)$ decays exponentially as $t \to \infty$ , that is, $\begin{matrix} (4.10) & E (t) ⩽ 3 E (0) e^{- \frac{2 ω}{3} t}, \forall t ⩾ 0 . \end{matrix}$ Moreover, if the source term $f (u) = 0$ , then the exponential decay (4.10) holds for $E (t)$ given by (2.21). This is in agreement with the results from Magaña and Quintanilla [10, Theorem 2.2] in the sense that the same decay rate for the solutions of the linear system with quasi-static microvoids was obtained.
Proof of Theorem 4.8.
For an arbitrary $ω > 0$ , define the perturbed energy $\begin{matrix} (4.11) & V_{ω} (t) = E_{J} (t) + ω Φ (t), \end{matrix}$ where Φ is the functional $\begin{matrix} Φ (t) = ρ ⟨ u, u_{t} ⟩ + J ⟨ ϕ, ϕ_{t} ⟩ + \frac{1}{2} (γ ‖ u_{x} ‖^{2} + τ ‖ ϕ ‖^{2}) . \end{matrix}$ We first claim that there exists is a constant $ω_{0} > 0$ independent of $0 < J < 1$ such that $\begin{matrix} (4.12) & \frac{1}{2} E_{J} (t) - \frac{1}{2} C_{F} L ⩽ V_{ω} (t) ⩽ \frac{3}{2} E_{J} (t) + \frac{1}{2} C_{F} L, \forall ω \in [0, ω_{0}] . \end{matrix}$ Indeed, using Young’s inequality, (2.3) and (2.11), we have $\begin{matrix} | Φ (t) | & ⩽ ρ ‖ u ‖ ‖ u_{t} ‖ + J ‖ ϕ ‖ ‖ ϕ_{t} ‖ + \frac{1}{2} (γ ‖ u_{x} ‖^{2} + τ ‖ ϕ ‖^{2}) \\ ⩽ \frac{1}{2} (ρ ‖ u_{t} ‖^{2} + J ‖ ϕ_{t} ‖^{2}) + C (‖ u_{x} ‖^{2} + ‖ ϕ_{x} ‖^{2}) \\ ⩽ C {‖ y (t) ‖}_{H}^{2} ⩽ C (E_{J} (t) + C_{F} L), \end{matrix}$ with $C > 0$ independent of $0 < J < 1$ . Therefore, taking $ω_{0} = \frac{1}{2 C}$ , we conclude that (4.12) holds.

Now, multiplying the equations in (1.6) by u and ϕ, respectively, we find that $\begin{matrix} \frac{d}{d t} Φ (t) = ρ ‖ u_{t} ‖^{2} + J ‖ ϕ_{t} ‖^{2} - μ ‖ u_{x} ‖^{2} - δ ‖ ϕ_{x} ‖^{2} - ξ ‖ ϕ ‖^{2} - 2 b ⟨ u_{x}, ϕ ⟩ - \int_{0}^{L} f (u) u d x . \end{matrix}$ Subtracting and adding $E_{J} (t)$ , we get $\begin{matrix} (4.13) & \begin{matrix} \frac{d}{d t} Φ (t) & = - E_{J} (t) + \frac{3 ρ}{2} ‖ u_{t} ‖^{2} + \frac{3 J}{2} ‖ ϕ_{t} ‖^{2} - \frac{1}{2} (μ ‖ u_{x} ‖^{2} + δ ‖ ϕ_{x} ‖^{2} + ξ ‖ ϕ ‖^{2} + 2 b ⟨ u_{x}, ϕ ⟩) \\ + \int_{0}^{L} (F (u) - f (u) u) d x . \end{matrix} \end{matrix}$ Using the assumption (2.6) and Poincaré’s inequality, we find that $\begin{matrix} (4.14) & \frac{d}{d t} Φ (t) ⩽ - E_{J} (t) + \frac{3 ρ}{2 λ_{1}} ‖ u_{x t} ‖^{2} + \frac{3 J}{2} ‖ ϕ_{t} ‖^{2} + C_{F} L, \end{matrix}$ where $λ_{1} > 0$ is the Poincaré’s constant. Using (2.10), (4.14), we obtain $\begin{matrix} \frac{d}{d t} V_{ω} (t) ⩽ - ω E_{J} (t) - (γ - \frac{3 ρ ω}{2 λ_{1}}) ‖ u_{x t} ‖^{2} - (τ - \frac{3 J ω}{2}) ‖ ϕ_{t} ‖^{2} + ω C_{F} L . \end{matrix}$ Thus, taking $\begin{matrix} ω = min {ω_{0}, \frac{2 λ_{1} γ}{3 ρ}, \frac{2 τ}{3 J ω}}, \end{matrix}$ we conclude that $\begin{matrix} \frac{d}{d t} V_{ω} (t) ⩽ - ω E_{J} (t) + ω C_{F} L . \end{matrix}$ On account of second inequality in (4.12), we have $\begin{matrix} \frac{d}{d t} V_{ω} (t) ⩽ - \frac{2 ω}{3} V_{ω} (t) + \frac{4}{3} ω C_{F} L . \end{matrix}$ Thus, the Gronwall’s lemma yields $\begin{matrix} V_{ω} (t) ⩽ V_{ω} (0) e^{- \frac{2 ω}{3} t} + 2 C_{F} L . \end{matrix}$ By using (4.12) again, we arrive at (4.8). The proof is complete. □
Remark 4.10.
Combining the stability inequality (4.8) with energy estimate (2.11), we see that given $y_{0} \in H$ , $\begin{matrix} (4.15) & {‖ S_{J} (t) y_{0} ‖}_{H_{J}}^{2} ⩽ 2 (3 C_{F} ‖ y_{0} ‖_{H_{J}}^{p + 1} + C_{F} (L + 3)) e^{- \frac{2 ω}{3} t} + 12 C_{F} L . \end{matrix}$ Therefore, any closed ball $B = B (0, R)$ in $H$ of center zero and radius $R > 12 C_{F} L$ is a absorbing set absorbing set for the system $(H, S_{J} (t))$ uniformly with respect to $0 < J < 1$ .
Remark 4.11.
As a consequence of inequality (4.9), we see that the $B_{0} = B (0, R_{0})$ in $H_{0}$ of center zero and radius $R_{0} > 12 C_{F} L$ is a absorbing for the system $(H_{0}, S (t))$ .

4.3. Uniform quasi-stability as $J \to 0$

In this subsection, we prove the uniform quasi-stability for the dynamical system $(H, S_{J} (t))$ on any forward-invariant bounded set B of $H$ . We get the quasi-stability on B by establishing a observability inequality uniformly in $0 < J < 1$ . We also prove the quasi-stability for the dynamical system $(H_{0}, S (t))$ as limit of the uniform quasi-stability of the dynamical system $(H, S_{J} (t))$ as $J \to 0$ .

Theorem 4.12.
The dynamical system $(H, S_{J} (t))$ corresponding to problem ( 1.6 )–( 1.8 ) is quasi-stable on any forward-invariant bounded set of $H$ uniformly with respect to $0 < J < 1$ .
Proof.
We start by considering the difference of two strong solutions $y^{i} (t) = (u^{i} (t), ϕ^{i} (t), u_{t}^{i} (t), ϕ_{t}^{i} (t))$ to (1.6), $i = 1, 2$ . Thus, taking the notation $\begin{matrix} u = u^{1} - u^{2}, ϕ = ϕ^{1} - ϕ^{2}, F (u) = f (u^{1}) - f (u^{2}), \end{matrix}$ we obtain the system $\begin{matrix} (4.16) & \{\begin{matrix} ρ u_{t t} - μ u_{x x} - b ϕ_{x} - γ u_{x x t} + F (u) = 0, & in (0, L) \times (0, \infty), \\ J ϕ_{t t} - δ ϕ_{x x} + b u_{x} + ξ ϕ + τ ϕ_{t} = 0, & in (0, L) \times (0, \infty), \end{matrix} \end{matrix}$ with boundary conditions $\begin{matrix} (4.17) & u (0, t) = u (L, t) = ϕ_{x} (0, t) = ϕ_{x} (L, t) = 0, t ⩾ 0, \end{matrix}$ and initial condition $\begin{matrix} (4.18) & \begin{matrix} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), \\ ϕ (x, 0) = ϕ_{0} (x), ϕ_{t} (x, 0) = ϕ_{1} (x), in (0, L) . \end{matrix} \end{matrix}$ Now, on every bounded positively invariant set B of $H$ , we have $\begin{matrix} sup_{t ⩾ 0} ({‖ y^{1} (t) ‖}_{H_{J}}^{2} + {‖ y^{2} (t) ‖}_{H_{J}}^{2}) ⩽ C_{B} . \end{matrix}$ Multiplying the equations in (4.16) by u and ϕ, respectively, yields $\begin{matrix} (4.19) & \begin{matrix} \int_{0}^{T} E_{J} (t) d t & = - \frac{1}{2} {[ρ ⟨ u_{t}, u ⟩ + J ⟨ ϕ_{t}, ϕ ⟩]}_{0}^{T} + \int_{0}^{T} (ρ ‖ u_{t} ‖^{2} + J ‖ ϕ_{t} ‖^{2}) d t \\ - \frac{1}{2} \int_{0}^{T} (γ ⟨ u_{x t}, u_{x} ⟩ + τ ⟨ ϕ_{t}, ϕ ⟩) d t - \frac{1}{2} \int_{0}^{T} ⟨ F (u), u ⟩ d t . \end{matrix} \end{matrix}$ Let us estimate the right side of (4.19). Using Hölder and Young inequalities and (2.4), we deduce that $\begin{matrix} ρ ⟨ u_{t}, u ⟩ + J ⟨ ϕ_{t}, ϕ ⟩ ⩽ \frac{1}{2} (ρ^{2} ‖ u_{t} ‖^{2} + J^{2} ‖ ϕ_{t} ‖^{2}) + \frac{1}{2} (‖ u ‖^{2} + ‖ ϕ ‖^{2}) ⩽ C E_{J} (t), \end{matrix}$ for some constante $C > 0$ independent of $0 < J < 1$ . Thus, there exists a constant $C > 0$ , independent of $0 < J < 1$ , such that $\begin{matrix} (4.20) & {[ρ_{1} ⟨ φ_{t}, φ ⟩ + ρ_{2} ⟨ ψ_{t}, ψ ⟩]}_{0}^{T} ⩽ C (E_{J} (0) + E_{J} (T)) . \end{matrix}$ Using Young’s inequality, we find for any $ϵ > 0$ , $\begin{matrix} (4.21) & \begin{matrix} \int_{0}^{T} (γ ⟨ u_{x t}, u_{x} ⟩ + τ ⟨ ϕ_{t}, ϕ ⟩) d t & ⩽ C \int_{0}^{T} (γ ‖ u_{x t} ‖ ‖ u_{x} ‖ + τ ‖ ϕ_{t} ‖ ‖ ϕ ‖) d t \\ ⩽ C_{γ, τ} \int_{0}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t + ϵ \int_{0}^{T} E_{J} (t) d t . \end{matrix} \end{matrix}$ Using (2.5) and the embedding $H^{1} (0, L) ↪ L^{\infty} (0, L)$ , we obtain that $\begin{matrix} (4.22) & \int_{0}^{T} ⟨ F (u), u ⟩ d t ⩽ C_{B} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$ Inserting the estimates (4.20)–(4.22) into (4.19) and choosing $ϵ > 0$ small enough, we conclude that there exist constants $C > 0$ and $C_{B} > 0$ independently of $0 < J < 1$ such that $\begin{matrix} \int_{0}^{T} E_{J} (t) d t & ⩽ C (E_{J} (T) + E_{J} (0)) + C_{γ, τ} \int_{0}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t \\ + \int_{0}^{T} (ρ ‖ u_{t} ‖^{2} + J ‖ ϕ_{t} ‖^{2}) d t + C_{B} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$ Using the Poincaré’s inequality, we deduce that $\begin{matrix} (4.23) & \int_{0}^{T} E_{J} (t) d t ⩽ C (E_{J} (T) + E_{J} (0)) + C_{γ, τ} \int_{0}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t + C_{B} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$

Next, we use the multiplies $u_{t}$ and $ϕ_{t}$ in (4.16) to obtain $\begin{matrix} (4.24) & \int_{s}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t = E_{J} (s) - E_{J} (T) - \int_{s}^{T} ⟨ F (u), u_{t} ⟩ d t . \end{matrix}$ Using (2.5), we have for any $ϵ > 0$ , $\begin{matrix} (4.25) & \int_{0}^{T} ⟨ F (u), u_{t} ⟩ d t ⩽ C_{B} \int_{0}^{T} ‖ u ‖ ‖ u_{t} ‖ d t ⩽ C_{B} \int_{0}^{T} ‖ u ‖^{2} d t + ϵ \int_{0}^{T} E_{J} (t) d t . \end{matrix}$ Now use (4.24) and (4.25) to get $\begin{matrix} (4.26) & \int_{0}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t ⩽ E_{J} (0) - E_{J} (T) + ϵ \int_{0}^{T} E_{J} (t) d t + C_{B} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$ Next, we combine estimates (4.23) and (4.26) for $ϵ > 0$ small enough to obtain $\begin{matrix} (4.27) & \int_{0}^{T} E_{J} (t) d t ⩽ (C - C_{γ, τ}) E_{J} (T) + (C + C_{γ, τ}) E_{J} (0) + C_{B} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$ Now, integrate the energy equality (4.24) with respect to s so that $\begin{matrix} T E_{J} (T) = \int_{0}^{T} E_{J} (t) d t - \int_{0}^{T} \int_{s}^{T} (γ ‖ u_{x t} ‖^{2} + τ ‖ ϕ_{t} ‖^{2}) d t d s - \int_{0}^{T} \int_{s}^{T} ⟨ F (u), u_{t} ⟩ d t d s . \end{matrix}$ Thus, by estimate (4.25), the following is immediate $\begin{matrix} (4.28) & T E_{J} (T) ⩽ 2 \int_{0}^{T} E_{J} (t) d t + C_{B, T} \int_{0}^{T} ‖ u ‖^{2} d t . \end{matrix}$ Inserting the estimate (4.27) into (4.28) yields $\begin{matrix} T E_{J} (T) ⩽ 2 (C - C_{γ, τ}) E_{J} (T) + 2 (C + C_{γ, τ}) E_{J} (0) + C_{B, T} \int_{0}^{T} ‖ u ‖^{2} d t, \end{matrix}$ for some constants $C, C_{γ, τ}, C_{B, T} > 0$ independent of $0 < J < 1$ . We choose $T > 4 C$ to deduce that $\begin{matrix} (4.29) & E_{J} (T) ⩽ ς_{T} E_{J} (0) + C_{B, T} sup_{s \in [0, T]} {‖ u (s) ‖}^{2}, \end{matrix}$ where $\begin{matrix} ς_{T} = \frac{2 (C + C_{γ, τ})}{T - 2 (C - C_{γ, τ})} < 1 . \end{matrix}$ By using a standard argument as in [2, p.100]), we conclude that there exist $ς_{B}, ω_{B}, C_{B} > 0$ independent of $0 < J < 1$ such that $\begin{matrix} (4.30) & E_{J} (t) ⩽ ς_{B} E_{J} (0) e^{- ω_{B} t} + C_{B} sup_{s \in [0, t]} {‖ u (s) ‖}^{2}, \end{matrix}$ and thereby $(H, S_{J} (t))$ is quasi-stable on B uniformly with respect to $0 < J < 1$ . The proof is complete. □
Remark 4.13.
As a limit of (4.30) when $J \to 0$ , we obtain the following quasi-stability estimate for the dynamical system $(H_{0}, S (t))$ : For any bounded positively invariant set $B_{0}$ of $H_{0}$ , there exist $ς_{B_{0}}, ω_{B_{0}}, C_{B_{0}} > 0$ such that $\begin{matrix} (4.31) & E (t) ⩽ ς_{B_{0}} E (0) e^{- ω_{B_{0}} t} + C_{B_{0}} sup_{s \in [0, t]} {‖ u (s) ‖}^{2} . \end{matrix}$

4.4. Proof of Theorem 4.6

(1) From Theorem 4.12 the system $(H, S_{J} (t))$ is quasi-stable and consequently it is asymptotically smooth by Theorem 4.3. Thus, since $(H, S_{J} (t))$ is also dissipative by Lemma 4.10, the existence of a unique compact global attractor $A_{J}$ is established by Theorem 4.1.

(2) From Theorem 4.5(i), we know that the attractor $A_{J}$ has finite fractal dimension. Since the quasi-stability is uniform in $0 < J < 1$ , Remark 7.9.7 in [3] shows that the finite fractal dimension is independent on $0 < J < 1$ .

(3) Let $y (t) = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t))$ be a full trajectory in $A_{J}$ . Since the bounded absorbing set $B$ given in Remark 4.10 is uniform with respect to J we conclude that $\begin{matrix} (4.32) & sup_{t \in R} {‖ y (t) ‖}_{H_{J}}^{2} ⩽ R, \forall t \in R, \end{matrix}$ for some constant $R > 0$ independent of $0 < J < 1$ . Thus, the uniform estimate of the sup in (4.6) is obtained. The finiteness of the dissipation integral in (4.6) follows by (2.16).

Now, since the coeficiente $c (t) = C_{B}$ in (4.30) is independent of $0 < J < 1$ , by Theorem 4.5(ii) there exists a constant $R > 0$ independent of $0 < J < 1$ such that $\begin{matrix} (4.33) & sup_{t \in R} {‖ y_{t} (t) ‖}_{H_{J}}^{2} ⩽ R, \forall t \in R . \end{matrix}$ Using the equations in (1.6), we have $\begin{matrix} (4.34) & \{\begin{matrix} {(μ u + γ u_{t})}_{x x} = ρ u_{t t} - b ϕ_{x} + f (u), \\ δ ϕ_{x x} = J ϕ_{t t} + b u_{x} + ξ ϕ + τ ϕ_{t} . \end{matrix} \end{matrix}$ Thus, using the first equation in (4.34), uniform estimates (4.32)–(4.33) and the fact that $f (u)$ is locally Lipschitz continuous, we conclude that $\begin{matrix} (4.35) & sup_{t \in R} {‖ {(μ u (t) + γ u_{t} (t))}_{x x} ‖}^{2} ⩽ C sup_{t \in R} ({‖ u_{t t} (t) ‖}^{2} + {‖ ϕ_{x} (t) ‖}^{2} + {‖ f (u (t)) ‖}^{2}) ⩽ R, \end{matrix}$ with $R > 0$ independent of $0 < J < 1$ . On the other hand, second equality in (4.34) and estimates (4.32)–(4.33) yields that $\begin{matrix} (4.36) & sup_{t \in R} {‖ ϕ_{x x} (t) ‖}^{2} ⩽ C sup_{t \in R} (J^{2} {‖ ϕ_{t t} (t) ‖}^{2} + {‖ u_{x} (t) ‖}^{2} + {‖ ϕ_{t} (t) ‖}^{2}) ⩽ R, \end{matrix}$ with $R > 0$ independent of $0 < J < 1$ . Now, using (4.33), (4.35) and (4.36) we conclude that (4.7) holds. Finally, since the global attractor is the set of all bounded full trajectories, we see that $A_{J}$ is bounded in $V$ . This ends the proof of Theorem 4.6. □

Concerning global attractors for the quasi-static problem (1.9)–(1.11) we have the following result.

Theorem 4.14.
Under the hypotheses of Theorem 4.6 , we have
Global attractors: The dynamical system $(H_{0}, S (t))$ possesses a global attractor $A$ .

Finite-dimensionality: The global attractor $A$ has finite fractal dimension ${dim}_{f}^{H_{0}} A$ .

Regularity: The global attractor $A$ is bounded in $\begin{matrix} V_{0} = H_{0}^{1} (0, L) \times H_{0}^{1} (0, L) \times H_{}^{2} (0, L) . \end{matrix}$ Moreover, every full trajectory* $z (t) = (u (t), u_{t} (t), ϕ (t))$ from the attractor satisfies $\begin{matrix} (4.37) & sup_{t \in R} ({‖ u_{t} (t) ‖}^{2} + {‖ u_{x} (t) ‖}^{2} + {‖ ϕ_{x} (t) ‖}^{2} + \int_{- \infty}^{\infty} {‖ u_{x t} (s) ‖}^{2} d s) ⩽ R, \end{matrix}$ and $\begin{matrix} (4.38) & sup_{t \in R} ({‖ {(μ u (t) + γ u_{t} (t))}_{x x} ‖}^{2} + {‖ ϕ_{x x} (t) ‖}^{2} + {‖ u_{t t} (t) ‖}^{2} + {‖ u_{x t} (t) ‖}^{2} + {‖ ϕ_{x t} (t) ‖}^{2}) ⩽ R, \end{matrix}$ for some constant $R > 0$ .

Proof.
The argument is the same as in the proof of Theorem 4.6 by taking into account the Remarks 4.11 and 4.13. The proof is complete. □

5. Upper semicontinuity of the attractor as $J \to 0$

In this section, we are interested in the convergence of the attractor $A_{J}$ of the system of the porous-elasticity (1.6) to the attractor of the quasi-static system (1.9) when $J \to 0$ . The main result of this section reads as follows.

Theorem 5.1.

The global attractor $A_{J}$ is upper semicontinuous as $J \to 0$ , that is, $\begin{matrix} (5.1) & lim_{J \to 0} {dist}_{H_{η}} (A_{J}, A_{0}) = 0, \forall η > 0, \end{matrix}$ where $H_{η} = H_{0}^{1 - η} (0, L) \times H_{*}^{1} (0, L) \times H_{0}^{1 - η} (0, L) \times H^{1 - η} (0, L)$ and $\begin{matrix} A_{0} = \{(u, ϕ, \tilde{u}, \tilde{ϕ}) | \begin{matrix} (u, \tilde{u}, ϕ) \in A \\ \tilde{ϕ} = τ^{- 1} (δ ϕ_{x x} - b u_{x} - ξ ϕ) \end{matrix}\} . \end{matrix}$ Here, $A$ is the global attractor for the dynamical system $(H_{0}, S (t))$ generated by the quasi-static problem ( 1.9 )–( 1.11 ).

Proof.

As in [5,6] we apply standard contradiction argument. Indeed, assume that (5.1) is not valid. Then there exist sequences $J_{n} \to 0$ and $y_{0}^{n} \in A_{J_{n}}$ such that $\begin{matrix} (5.2) & {dist}_{H_{η}} (y_{0}^{n}, A_{0}) ⩾ ϵ > 0, \forall n . \end{matrix}$ Let $y^{n} (t) = (u^{n} (t), ϕ^{n} (t), u_{t}^{n} (t), ϕ_{t}^{n} (t))$ be a full trajectory in the attractor $A_{J_{n}}$ such that $y^{n} (0) = y_{0}^{n}$ . It follows from (4.6), (4.7) and Aubin–Lions theorem (see, e.g. [16], Corollary 4) that the sequence $y^{n}$ is relatively compact in the space $C ([- T, T]; H_{η})$ for every $T > 0$ . Thus, there exists $\begin{matrix} y (t) = (u (t), ϕ (t), u_{t} (t), ϕ_{t} (t)) \in C ([- T, T]; H_{η}) . \end{matrix}$ such that, up to a subsequence, $\begin{matrix} (5.3) & lim_{n \to \infty} max_{t \in [- T, T]} {‖ y^{n} (t) - y (t) ‖}_{H_{η}} = 0 . \end{matrix}$ Moreover, using (4.6), we deduce that $\begin{matrix} (5.4) & \begin{matrix} u_{t}^{n} \to u_{t} weakly in L^{2} (R; H_{0}^{1} (0, L)), \\ ϕ_{t}^{n} \to ϕ_{t} weakly in L^{2} (R; L^{2} (0, L)) . \end{matrix} \end{matrix}$ It follows from estimate (4.7) that $\begin{matrix} (5.5) & sup_{t \in R} (J_{n} ‖ ϕ_{t t}^{n} (t) ‖) = \sqrt{J_{n}} sup_{t \in R} (\sqrt{J_{n}} ‖ ϕ_{t t}^{n} (t) ‖) ⩽ \sqrt{J_{n}} R \to 0 as n \to \infty . \end{matrix}$ Using the same argument as in Theorem 3.1, we can pass to the limit in the variational form of problem (1.6)–(1.8) to see that $(u, ϕ)$ is a strong solution to quasi-static problem (1.9). Moreover, uniform estimates in (4.6) and (4.7) imply that $\begin{matrix} sup_{t \in R} ({‖ u_{x} (t) ‖}^{2} + {‖ u_{t} (t) ‖}^{2} + {‖ ϕ_{x} (t) ‖}^{2}) < \infty . \end{matrix}$ It follows that $(u (t), u_{t} (t), ϕ (t))$ is a bounded full trajectory for the system (1.9). Thus, $(u (0), u_{t} (0), ϕ (0)) \in A$ and since $\begin{matrix} ϕ_{t} (0) = τ^{- 1} (δ ϕ_{x x} (0) - b u_{x} (0) - ξ ϕ (0)), \end{matrix}$ we deduce by (5.3) that $\begin{matrix} y^{n} (0) = (u^{n} (0), ϕ^{n} (0), u_{t}^{n} (0), ϕ_{t}^{n} (0)) \to (u (0), ϕ (0), u_{t} (0), ϕ_{t} (0)) \in A_{0} in H_{η}, \end{matrix}$ which is impossible by (5.2). The proof is complete now. □

Footnotes

Acknowledgements

M.M. Freitas thank the CNPq for financial support by Grant 313081/2021-2. M.L. Santos wants to thank CNPq for financial support through the projects: CNPq Grant 308056/2021-3. A.J.A. Ramos thank the CNPq for financial support by Grant 310729/2019-0.

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