Global existence,asymptotic behavior and uniform attractor for a non-autonomous Timoshenko system of type III with weak damping

Abstract

In this paper, we investigate one-dimensional thermoelastic system of Timoshenko type III with double memory dampings. At first we give the global existence of solutions by using semigroup theory. Then we can prove the energy decay of solutions by constructing a series of Lyapunov functionals and obtain the existence of absorbing ball. Finally, we prove the asymptotic compactness by using uniform contractive functions and obtain the existence of uniform attractor.

Keywords

Timoshenko system global existence asymptotic behavior uniform attractor

1. Introduction

It is well-known that Timoshenko [29] developed a simple model to describe the transverse vibration of a beam. The system of coupled hyperbolic equations is given by $\begin{matrix} (1.1) & \{\begin{matrix} ρ u_{t t} = {(K (u_{x} - φ))}_{x}, & in (0, L) \times (0, + \infty), \\ I_{ρ} φ_{t t} = {(E I φ_{x})}_{x} + K (u_{t} - φ), & in (0, L) \times (0, + \infty) . \end{matrix} \end{matrix}$

Here, t represents the time and x denotes the space variable along the beam of length L, u is the transverse displacement of the beam from its equilibrium configuration and φ is the rotational angle of the filament of the beam. The coefficients ρ, $I_{ρ}$ , E, I and K represent the mass density, the polar moment of inertia of a cross-section, the Young’s modulus of elasticity, the moment of inertia of a cross-section, and the shear modulus, respectively.

For almost a century, a great number of researchers got interested in studying (1.1). In fact, the stability for a dissipative Timoshenko system with only one locally distributed damping depends on if the waves propagation are equal. This has been demonstrated by Soufyane and Wehbe [27], Messaoudi and Mustafa [13], Muñoz Rivera and Sare [17] and others. Ammar-Khodja et al. [1] considered a linear Timoshenko type system with a memory term and proved that the system is uniformly stable if and only if the wave speeds are equal $(\frac{K}{ρ_{1}} = \frac{b}{ρ_{2}})$ and g decays uniformly. For more regarding Timoshenko systems with memory term, we refer to Guesmia and Messaoudi [9], Keddi, et al. [10], Apalara [2], Messaoudi and Hassan [12], Messaoudi and Houari [16].

In the presence of two internal feedback controls, Raposo, et al. [26] studied the exponential decay of the solution of a linear Timoshenko type beam equation with frictional dissipative terms. Precisely, they studied the following system $\begin{matrix} \{\begin{matrix} ρ_{1} u_{t t} - K {(u_{x} - ψ)}_{x} + u_{t} = 0, & in (0, L) \times (0, + \infty), \\ ρ_{2} ψ_{t t} - b ψ_{x x} + K (u_{x} - ψ) + ψ_{t} = 0, & in (0, L) \times (0, + \infty), \\ u (0, t) = u (L, t) = ψ (0, t) = ψ (L, t) = 0, & t > 0 . \end{matrix} \end{matrix}$ and used the semigroup method developed by Liu and Zheng [11] to prove the exponential decay of the solution of the above system.

Recently, Qin and Wei [25] established the global existence and asymptotic behavior of solutions by using the semigroup method and multiplicatives, then proved the existence of a uniform attractor for a non-autonomous thermoelastic system by using the method of uniform contractive functions. Qin and Ma [24] used the multiplier techniques to prove the stability of the Timoshenko system of the type I classical thermoelasticity. In addition, the existence of the global attractor is obtained. For more relevant methods and results, we refer to [14,15,18–23].

Ghennam and Djebabla [8] considered a thermoelastic system (with $f = 0$ , $g = 0$ , $h = 0$ ) where the oscillations are defined by the Timoshenko model and the heat conduction is given by Greem and Naghdi theories. They introduced two new stability numbers and proved a general decay result, from which the exponential and polynomial decays are only special cases.

In the present work, we consider the following nonhomogeneous system $\begin{matrix} (1.2) & \{\begin{matrix} ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + γ θ_{t x} + μ φ_{t} = f (x, t), \\ ρ_{2} ψ_{t t} - b_{1} ψ_{x x} + k (φ_{x} + ψ) - γ θ_{t} + \int_{0}^{t} g_{1} (t - s) ψ_{x x} (x, s) d s = g (x, t), \\ ρ_{3} θ_{t t} - b_{2} θ_{x x} + \int_{0}^{t} g_{2} (t - s) θ_{x x} (x, s) d s + γ φ_{t x} + γ ψ_{t} = h (x, t), \end{matrix} \end{matrix}$ in $(x, t) \in Ω \times ℝ_{+}$ , where $Ω = (0, L)$ . The coefficients $ρ_{1}$ , $ρ_{2}$ , $ρ_{3}$ , k, $b_{1}$ , $b_{2}$ , γ and μ are positive constants, $μ φ_{t}$ represents a frictional damping, $\int_{0}^{t} g_{1} (t - s) ψ_{x x} (x, s) d s$ and $\int_{0}^{t} g_{2} (t - s) θ_{x x} (x, s) d s$ represent memory dampings, f, g, h are forcing terms.

We endow (1.2) with boundary conditions $\begin{matrix} (1.3) & φ (0, t) = φ (L, t) = ψ (0, t) = ψ (L, t) = θ (0, t) = θ (L, t) = 0, \forall t ⩾ 0 . \end{matrix}$ The initial data are given by $\begin{matrix} (1.4) & \{\begin{matrix} (φ (x, 0), ψ (x, 0), θ (x, 0)) = (φ_{0} (x), ψ_{0} (x), θ_{0} (x)), & \forall x \in Ω, \\ (φ_{t} (x, 0), ψ_{t} (x, 0), θ_{t} (x, 0)) = (φ_{1} (x), ψ_{1} (x), θ_{1} (x)), & \forall x \in Ω . \end{matrix} \end{matrix}$

To our knowledge, different damping mechanisms can be used to stabilize the vibration of the system and result in a more accurate decay rate, so we consider the asymptotic behavior of the system when all three equations of (1.2) have damping terms. The paper is organized as follows. In Section 2, we present some assumptions and preliminary works. Then in Section 3, we shall use the semigroup method to prove an existence result. We also state and prove asymptotic behavior of solutions by using the multiplicative method and some arguments from Zheng [30] in Section 4. Moreover, in Section 5, we prove the existence of the uniform attractor.

2. Preliminaries

In this section, we introduce some materials needed in the proof of our results. Throughout this paper, we use $c_{0}$ to denote a positive generic constant. In order to deal with the memory term, we denote by ∗ the usual convolution term and by ◇ and ∘ the binary operators $\begin{array}{l} (2.1) & (f * h) (t) = \int_{0}^{t} f (t - s) h (s) d s, \\ (2.2) & (f ◇ h) = \int_{0}^{t} f (t - s) (h (t) - h (s)) d s, \\ (2.3) & (f \circ h) = \int_{0}^{t} f (t - s) {(h (t) - h (s))}^{2} d s . \end{array}$

For the relaxation functions $g_{i}$ ( $i = 1, 2$ ) satisfy

$g_{i} : ℝ_{+} \to ℝ_{+}$ are differentiable functions such that $\begin{matrix} g_{i} (0) > 0, λ_{i} = b_{i} - \int_{0}^{\infty} g_{i} (s) d s > 0, i = 1, 2 . \end{matrix}$

There exist non-increasing functions $ζ_{i} : ℝ_{+} \to ℝ_{+}$ satisfy $\begin{matrix} g_{i}^{'} (t) ⩽ - ζ_{i} (t) g_{i} (t), i = 1, 2, \forall t > 0 . \end{matrix}$

Lemma 2.1 (Fatori and Muñoz Rivera [6]).

Assume that the function g satisfies $(H_{1})$ , then for any $h \in L^{2} (0, L)$ , there exists a positive constant $c_{0}$ such that $\begin{matrix} \int_{Ω} (\int_{0}^{t} g (t - s) (h (t) - h {(s)) d s)}^{2} d x ⩽ c_{0} \int_{Ω} (g \circ h) d x, \forall t ⩾ 0 . \end{matrix}$

Remark 2.1.
Using Lemma 2.1 and Poincaré’s inequality, with $- g^{'}$ instead of g, we can obtain $\begin{matrix} \int_{Ω} (\int_{0}^{t} - g^{'} (t - s) (h (t) - h {(s)) d s)}^{2} d x ⩽ c_{0} \int_{Ω} (g^{'} \circ h_{x}) d x . \end{matrix}$
Lemma 2.2 (Ghennam and Djebabla [8]).

Assume that the function g satisfies $(H_{1})$ , then for any $h \in L^{2} (0, L)$ , there exists a positive constant $c_{0}$ such that $\begin{matrix} \int_{Ω} {(h_{x} - (g * h_{x}))}^{2} d x ⩽ c_{0} (\int_{Ω} h_{x}^{2} d x + \int_{Ω} (g \circ h_{x}) d x), \forall t ⩾ 0 . \end{matrix}$

3. Well-posedness

In order to state and prove our main result, we at first set the vector function $U (t) = {(φ, u, ψ, v, θ, ω)}^{T}$ , where $u = φ_{t}$ , $v = ψ_{t}$ , $ω = θ_{t}$ .

Then the system (1.2)–(1.4) is converted to the following abstract ODE $\begin{matrix} (3.1) & \{\begin{matrix} \frac{d U}{d t} + A U = F, t > 0, \\ U (0) = U_{0} = {(φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1})}^{T}, \end{matrix} \end{matrix}$ where operators $A$ and F are defined by $\begin{array}{l} A U = \{\begin{matrix} - φ_{t} \\ - \frac{k}{ρ_{1}} {(φ_{x} + ψ)}_{x} + \frac{γ}{ρ_{1}} θ_{t x} + \frac{μ}{ρ_{1}} φ_{t} \\ - ψ_{t} \\ - \frac{b_{1}}{ρ_{2}} ψ_{x x} + \frac{k}{ρ_{2}} (φ_{x} + ψ) - \frac{γ}{ρ_{2}} θ_{t} + \frac{1}{ρ_{2}} \int_{0}^{t} g_{1} (t - s) ψ_{x x} (x, s) d s \\ - θ_{t} \\ - \frac{b_{2}}{ρ_{3}} θ_{x x} + \frac{1}{ρ_{3}} \int_{0}^{t} g_{2} (t - s) θ_{x x} (x, s) d s + \frac{γ}{ρ_{3}} φ_{t x} + \frac{γ}{ρ_{3}} ψ_{t} \end{matrix}\}, \\ F = \{\begin{matrix} 0 \\ \frac{f}{ρ_{1}} \\ 0 \\ \frac{g}{ρ_{2}} \\ 0 \\ \frac{h}{ρ_{3}} \end{matrix}\} . \end{array}$ Then we define the energy space $\begin{matrix} H = {(H_{0}^{1} (0, 1) \times L^{2} (0, 1))}^{3}, \end{matrix}$ and the domain of $A$ $\begin{matrix} D (A) = {U \in H | φ, ψ, θ \in H_{0}^{1} (0, 1) \cap H^{2} (0, 1), u, v, ω \in H_{0}^{1} (0, 1)} . \end{matrix}$ Thus $A : D (A) \to H$ is a linear operator.

For $U (t) = {(φ, u, ψ, v, θ, ω)}^{T}$ and $\tilde{U} (t) = {(\tilde{φ,} \tilde{u}, \tilde{ψ,} \tilde{v}, \tilde{θ}, \tilde{ω})}^{T}$ , we have the inner product of $H$ $\begin{array}{rcl} {(U, \tilde{U})}_{H} & = & \int_{Ω} {ρ_{1} u \tilde{u} + ρ_{2} v \tilde{v} + ρ_{3} ω \tilde{ω} + k (φ_{x} + ψ) (\tilde{φ_{x}} + \tilde{ψ}) + (b_{1} - \int_{0}^{t} g_{1} (s) d s) ψ_{x} \tilde{ψ_{x}} \\ (3.2) & + (b_{2} - \int_{0}^{t} g_{2} (s) d s) θ_{x} \tilde{θ_{x}} + g_{1} \circ ψ_{x} + g_{2} \circ θ_{x}} d x . \end{array}$

Theorem 3.1.
Similarly as [ 7 ], we chose that $f (x, t), g (x, t), h (x, t) \in L^{2} ([0, + \infty), L^{2} (0, 1))$ , and assume that $g_{i}$ ( $i = 1, 2$ ) satisfy $(H_{1})$ and $(H_{2})$ . Then for any $U_{0} \in H$ , system ( 1.2 )–( 1.4 ) has a unique global solution $\begin{matrix} U \in C ([0, + \infty), H) . \end{matrix}$ Moreover, if $U_{0} \in D (A)$ , system ( 1.2 )–( 1.4 ) have a unique solution $\begin{matrix} U \in C ([0, + \infty), D (A) \cap C^{1} ([0, + \infty), H) . \end{matrix}$

In order to complete the proof of Theorem 3.1, we need the following lemmas. For an abstract initial value problem $\begin{matrix} (3.3) & \{\begin{matrix} \frac{d y}{d t} + A y = F (t), \\ y (0) = y_{0}, \end{matrix} \end{matrix}$ where $A$ is a maximal accretive operator defined in a dense subset $D (A)$ of Banach space $H$ . Lemma 3.1 (Zheng [30]).

Let $A$ be a linear operator defined in a Hilbert space $H, A : D (A) \subset H \to H$ . Then the necessary and sufficient conditions for $D (A)$ being maximal accretive are

$Re (A x, x) ⩽ 0$ , $\forall x \in D (A)$ .

$R (λ I - A) = H$ .

Lemma 3.2 (Zheng [30]).

Assume that $A$ is m-accretive in Banach space H, and $y_{0} \in D (A)$ , $F (t) \in C^{1} ([0, + \infty), H)$ . Then problem ( 3.3 ) has a unique classical solution $y (t)$ such that $\begin{matrix} y (t) \in C^{1} ([0, + \infty), H) \cap C ([0, + \infty), D (A)), \end{matrix}$ which can be expressed as $\begin{matrix} y (t) = S (t) y_{0} + \int_{0}^{t} S (t - τ) F (τ) d τ . \end{matrix}$

Proof of Theorem 3.1.
According to Lemma 3.1 (see also [7]), we can know that $A$ is an maximal monotone operator. By the assumptions, we have ${(φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1})}^{T} \in D (A)$ , and then by using Lemma 3.2, we complete the proof. □

4. Uniform stability

In this section, we consider system (1.2)–(1.4) and will state and prove our decay results. At first we introduce the multiplier $\begin{matrix} ω = - \int_{0}^{x} ψ (y, t) d y + \frac{x}{L} \int_{0}^{L} ψ (y, t) d y, \end{matrix}$ which satisfies $- ω_{x x} = ψ_{x}$ , $ω (0) = ω (L) = 0$ . Since the boundary condition for $ω_{t}$ can be derived from $ω (0, t) = ω (L, t) = 0$ , namely $ω_{t} (0, t) = ω_{t} (L, t) = ω (0, t) = ω (L, t) = 0$ . Then we can obtain, $\begin{matrix} ω_{t x} = - ψ_{t} (x, t) + \frac{1}{L} \int_{0}^{L} ψ_{t} (y, t) d y . \end{matrix}$ Squaring both sides of the above equation and integrating it, we can obtain $\begin{matrix} \int_{0}^{L} ω_{t x}^{2} d x ⩽ \int_{0}^{L} ψ_{t}^{2} d x + \int_{0}^{L} ψ_{t}^{2} d x . \end{matrix}$ Thus by using Poincaré’s inequality, we can obtain the following inequalities $\begin{array}{l} (4.1) & \int_{Ω} ω_{t}^{2} d x ⩽ C_{p} \int_{Ω} ω_{t x}^{2} d x ⩽ C_{p} \int_{Ω} ψ_{t}^{2} d x, \\ (4.2) & \int_{Ω} ω^{2} d x ⩽ C_{p} \int_{Ω} ω_{x}^{2} d x ⩽ C_{p} \int_{Ω} ψ^{2} d x ⩽ C_{p} \int_{Ω} ψ_{x}^{2} d x . \end{array}$ We define the functional energy of solutions of system (1.2)–(1.4) as follows $\begin{array}{l} E (t) = & \frac{1}{2} \int_{Ω} {ρ_{1} φ_{t}^{2} + ρ_{2} ψ_{t}^{2} + ρ_{3} θ_{t}^{2} + k {(φ_{x} + ψ)}^{2} + (b_{1} - \int_{0}^{t} g_{1} (s) d s) ψ_{x}^{2} \\ (4.3) & + (b_{2} - \int_{0}^{t} g_{2} (s) d s) θ_{x}^{2} + g_{1} \circ ψ_{x} + g_{2} \circ θ_{x}} d x . \end{array}$

Lemma 4.1.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then $E (t)$ satisfies $\begin{matrix} (4.4) & E^{'} (t) ⩽ \frac{1}{2} \int_{Ω} (g_{1}^{'} \circ ψ_{x} + g_{2}^{'} \circ θ_{x}) d x + ε \int_{0}^{L} (φ_{t}^{2} + ψ_{t}^{2} + θ_{t}^{2}) d x + C_{ε} \int_{0}^{L} (f^{2} + g^{2} + h^{2}) d x . \end{matrix}$
Proof.
Multiplying (1.2)₁, (1.2)₂, (1.2)₃ by $φ_{t}$ , $ψ_{t}$ , $θ_{t}$ respectively, then integrating them over Ω, we can get $\begin{array}{l} \frac{1}{2} \frac{d}{d t} E (t) = & - μ \int_{Ω} φ_{t}^{2} d x - \frac{1}{2} g_{1} (t) \int_{Ω} ψ_{x}^{2} d x - \frac{1}{2} g_{2} (t) \int_{Ω} θ_{x}^{2} d x + \frac{1}{2} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x \\ + \frac{1}{2} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x + \int_{Ω} f φ_{t} d x + \int_{Ω} g ψ_{t} d x + \int_{Ω} h θ_{t} d x \\ (4.5) & ⩽ & \frac{1}{2} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x + \frac{1}{2} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x + \int_{Ω} f φ_{t} d x + \int_{Ω} g ψ_{t} d x + \int_{Ω} h θ_{t} d x, \end{array}$ by using Young’s inequality and for any $ε > 0$ , we can obtain (4.4) easily. □
Theorem 4.1.
Let $((φ_{0}, φ_{1}), (ψ_{0}, ψ_{1}), (θ_{0}, θ_{1})) \in {(H_{0}^{1} (0, 1) \times L^{2} (0, 1))}^{3}$ be given, assume that hypotheses $(H_{1})$ , $(H_{2})$ hold, the coefficients $ρ_{1}$ , $ρ_{2}$ , $ρ_{3}$ , k, $b_{1}$ , $b_{2}$ , and γ satisfy the conditions $\begin{matrix} (4.6) & κ_{1} = γ + \frac{b_{1} ρ_{1}}{k} - ρ_{2} = 0, κ_{2} = \frac{ρ_{3} b_{1}}{ρ_{2}} + \frac{b_{1} γ}{k} - b_{2} = 0 . \end{matrix}$ $(φ, ψ, θ)$ is the solution of problem ( 1.2 )–( 1.4 ) and $f (x, t), g (x, t), h (x, t) \in L^{2} ([0, + \infty), L^{2} (0, 1))$ , then we have $\begin{matrix} lim_{t \to + \infty} E (t) = 0 . \end{matrix}$ If further, $‖ f ‖_{L^{2}}^{2} + ‖ g ‖_{L^{2}}^{2} + ‖ h ‖_{L^{2}}^{2} ⩽ C_{0} e^{- α_{0} t}$ , $\forall t > 0$ , with $C_{0} > 0$ and $α_{0} > 0$ being constants, then there exist positive constants M and α, such that the energy $E (t)$ satisfies $\begin{matrix} E (t) ⩽ M e^{- α t}, \forall t > 0 . \end{matrix}$ If $‖ f ‖_{L^{2}}^{2} + ‖ g ‖_{L^{2}}^{2} + ‖ h ‖_{L^{2}}^{2} ⩽ \frac{C^{'}}{{(1 + t)}^{p}}, \forall t > 0$ , with constants $C^{'} > 0$ , $p > 1$ , then there exist a constant $C^{} > 0$ such that* $\begin{matrix} E (t) ⩽ C^{} {(1 + t)}^{- p + 1}, \forall t > 0 . \end{matrix}$
Remark 4.1.
When $γ = 0$ , then $κ_{1} = 0$ , $κ_{2} = 0$ imply that $\frac{k}{ρ_{1}} = \frac{b_{1}}{ρ_{2}} = \frac{b_{2}}{ρ_{3}}$ , which is the usual case that we found in previous studies, we can refer to Soufyane and Wehbe [27], Ammar-Khodja et al. [1] and others.

In order to prove above results, we introduce various functionals and establish several lemmas.
Lemma 4.2.
Let* $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then the function $\begin{matrix} (4.7) & I_{1} (t) = ρ_{1} \int_{Ω} φ_{t} ω d x + ρ_{2} \int_{Ω} ψ_{t} ψ d x + μ \int_{Ω} φ ω d x, t ⩾ 0, \end{matrix}$ satisfies for any positive constant $ε_{1}$ , $\begin{array}{l} I_{1}^{'} (t) ⩽ & - \frac{λ_{1}}{2} \int_{Ω} ψ_{x}^{2} d x + c (ε_{1}) \int_{Ω} ψ_{t}^{2} d x + ε_{1} \int_{Ω} φ_{x}^{2} d x + ε_{1} \int_{Ω} φ_{t}^{2} d x \\ (4.8) & + c_{0} \int_{Ω} θ_{t}^{2} d x + c_{0} \int_{Ω} (g_{1} \circ ψ_{x}) d x + c_{0} \int_{Ω} (f^{2} + g^{2}) d x, \end{array}$ where $c (ε_{1}) = (ρ_{2} + \frac{ρ_{1}^{2} C_{p}}{2 ε_{1}} + \frac{μ^{2} C_{p}}{4 ε_{1}})$ .
Proof.
Differentiating $I_{1} (t)$ and using (1.2), we conclude that $\begin{array}{l} I_{1}^{'} (t) = & ρ_{2} \int_{Ω} ψ_{t}^{2} d x - k \int_{Ω} ψ^{2} d x + k \int_{Ω} ω_{x}^{2} d x - b_{1} \int_{Ω} ψ_{x}^{2} d x + ρ_{1} \int_{Ω} φ_{t} ω_{t} d x + γ \int_{Ω} θ_{t} ω_{x} d x \\ (4.9) & + γ \int_{Ω} θ_{t} ψ d x + \int_{Ω} (g_{1} * ψ_{x}) ψ_{x} d x + \int_{Ω} f ω d x + \int_{Ω} g ψ d x + μ \int_{Ω} φ ω_{t} d x . \end{array}$ Hence, by using the Young’s inequality, Poincaré’s inequality, Lemma 2.1, relations (4.1) and (4.2), we can arrive at the following estimates, for $ε_{1}, δ > 0$ , $\begin{array}{l} (4.10) & ρ_{1} \int_{Ω} φ_{t} ω_{t} d x ⩽ \frac{ε_{1}}{2} \int_{Ω} φ_{t}^{2} d x + \frac{ρ_{1}^{2} C_{p}}{2 ε_{1}} \int_{Ω} ψ_{t}^{2} d x, \\ (4.11) & γ \int_{Ω} θ_{t} ω_{x} d x ⩽ δ \int_{Ω} ψ_{x}^{2} d x + \frac{γ^{2} C_{p}}{4 δ} \int_{Ω} θ_{t}^{2} d x, \\ (4.12) & γ \int_{Ω} θ_{t} ψ d x ⩽ δ \int_{Ω} ψ_{x}^{2} d x + \frac{γ^{2} C_{p}}{4 δ} \int_{Ω} θ_{t}^{2} d x, \\ (4.13) & \int_{Ω} (g_{1} * ψ_{x}) ψ_{x} d x ⩽ (\int_{0}^{t} g_{1} (s) d s) \int_{Ω} ψ_{x}^{2} d x + δ \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{δ} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.14) & μ \int_{Ω} φ ω_{t} d x ⩽ ε_{1} \int_{Ω} φ_{x}^{2} d x + \frac{μ^{2} C_{p}}{4 ε_{1}} \int_{Ω} ψ_{t}^{2} d x, \\ (4.15) & \int_{Ω} f ω d x ⩽ \frac{C_{p}}{4 δ} \int_{Ω} f^{2} d x + δ \int_{Ω} ψ_{x}^{2} d x, \\ (4.16) & \int_{Ω} g ψ d x ⩽ \frac{C_{p}}{4 δ} \int_{Ω} g^{2} d x + δ \int_{Ω} ψ_{x}^{2} d x . \end{array}$ Substituting (4.10)–(4.16) into (4.9) and choose $δ = \frac{λ_{1}}{10}$ , then we can obtain (4.8). □
Lemma 4.3.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then the function $\begin{matrix} (4.17) & I_{2} (t) = - ρ_{1} \int_{Ω} φ_{t} φ d x - ρ_{2} \int_{Ω} ψ_{t} ψ d x - \frac{μ}{2} \int_{Ω} φ^{2} d x, t ⩾ 0, \end{matrix}$ satisfies the estimate $\begin{array}{l} I_{2}^{'} (t) ⩽ & - ρ_{1} \int_{Ω} φ_{t}^{2} d x - ρ_{2} \int_{Ω} ψ_{t}^{2} d x + \frac{5 k}{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + \frac{γ^{2}}{k} \int_{Ω} θ_{t}^{2} d x \\ (4.18) & + c_{0} \int_{Ω} ψ_{x}^{2} d x + c_{0} \int_{Ω} (g_{1} \circ ψ_{x}) d x + \frac{k}{4} \int_{Ω} φ_{x}^{2} d x + c_{0} \int_{Ω} f^{2} d x + c_{0} \int_{Ω} g^{2} d x . \end{array}$
Proof.
Differentiating $I_{2} (t)$ and using (1.2), we have $\begin{array}{l} I_{2}^{'} (t) = & - ρ_{1} \int_{Ω} φ_{t}^{2} d x - ρ_{2} \int_{Ω} ψ_{t}^{2} d x + k \int_{Ω} {(φ_{x} + ψ)}^{2} d x + (b_{1} - \int_{0}^{t} g_{1} (s) d s) \int_{Ω} ψ_{x}^{2} d x \\ (4.19) & + \int_{Ω} (g_{1} ◇ ψ_{x}) ψ_{x} d x - γ \int_{Ω} θ_{t} (φ_{x} + ψ) d x - \int_{Ω} f φ d x - \int_{Ω} g ψ d x . \end{array}$ Using Young’s inequality, Poincaré’s inequality and Lemma 2.1, we get $\begin{array}{l} (4.20) & - γ \int_{Ω} θ_{t} (φ_{x} + ψ) d x ⩽ \frac{γ^{2}}{k} \int_{Ω} θ_{t}^{2} d x + \frac{k}{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x, \\ \int_{Ω} (g_{1} ◇ ψ_{x}) ψ_{x} d x ⩽ \frac{λ_{1}}{2} \int_{Ω} ψ_{x}^{2} d x + \frac{1}{2 λ_{1}} \int_{Ω} {(g_{1} ◇ ψ_{x})}^{2} d x \\ (4.21) & ⩽ \frac{λ_{1}}{2} \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{2 λ_{1}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.22) & - \int_{Ω} f φ d x ⩽ \frac{C_{p}}{k} \int_{Ω} f^{2} d x + \frac{k}{4} \int_{Ω} φ_{x}^{2} d x, \\ (4.23) & - \int_{Ω} g ψ d x ⩽ \frac{C_{p}}{2 λ_{1}} \int_{Ω} g^{2} d x + \frac{λ_{1}}{2} \int_{Ω} ψ_{x}^{2} d x . \end{array}$ Combining (4.20)–(4.23) into (4.19), we can obtain (4.18). □
Lemma 4.4.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then the function $\begin{matrix} (4.24) & I_{3} = - ρ_{2} \int_{Ω} ψ_{t} (g_{1} ◇ ψ) d x, t ⩾ 0, \end{matrix}$ satisfies for any positive constant $ε_{2}$ and $δ_{1} > 0$ , $\begin{array}{l} I_{3}^{'} (t) ⩽ & - (ρ_{2} \int_{0}^{t} g_{1} (s) d s - δ_{1}) \int_{Ω} ψ_{t}^{2} d x - c_{0} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x + c_{0} ε_{2} \int_{Ω} ψ_{x}^{2} d x + ε_{2} \int_{Ω} θ_{t}^{2} d x \\ (4.25) & + ε_{2} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + c_{0} (ε_{2} + \frac{1}{ε_{2}}) \int_{Ω} (g_{1} \circ ψ_{x}) d x + C_{ε_{2}} \int_{Ω} g^{2} d x . \end{array}$
Proof.
Differentiating $I_{3} (t)$ and using (1.2), we have $\begin{array}{l} I_{3}^{'} (t) = & - (ρ_{3} \int_{0}^{t} g_{1} (s) d s) \int_{Ω} ψ_{t}^{2} d x - ρ_{2} \int_{Ω} ψ_{t} (g_{1}^{'} ◇ ψ) d x \\ + b_{1} \int_{Ω} (g_{1} ◇ ψ_{x}) ψ_{x} d x - γ \int_{Ω} θ_{t} (g_{1} ◇ ψ) d x \\ (4.26) & + k \int_{Ω} (φ_{x} + ψ) (g_{1} ◇ ψ) d x - \int_{Ω} (g_{1} * ψ_{x}) (g_{1} ◇ ψ_{x}) d x - \int_{Ω} g (g_{1} ◇ ψ) d x . \end{array}$ Using Young’s inequality, Poincaré’s inequality and Remark 2.1, we can conclude for all $δ_{1} > 0$ , $\begin{array}{l} - ρ_{2} \int_{Ω} ψ_{t} (g_{1}^{'} ◇ ψ) d x & ⩽ δ_{1} \int_{Ω} ψ_{t}^{2} d x + \frac{ρ_{2}^{2}}{4 δ_{1}} (\int_{0}^{t} - g_{1}^{'} (s) d s) \int_{Ω} (- g_{1}^{'} \circ ψ) d x \\ (4.27) & ⩽ δ_{1} \int_{Ω} ψ_{t}^{2} d x - \frac{c_{0}}{δ_{1}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x . \end{array}$ Similarly for any $ε_{2} > 0$ , we have $\begin{array}{l} (4.28) & b_{1} \int_{Ω} (g_{1} ◇ ψ_{x}) ψ_{x} d x ⩽ ε_{2} \int_{Ω} ψ_{x}^{2} d x + \frac{b_{1}^{2} c_{0}}{4 ε_{2}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.29) & - γ \int_{Ω} θ_{t} (g_{1} ◇ ψ) d x ⩽ ε_{2} \int_{Ω} θ_{t}^{2} d x + \frac{γ^{2} C_{p}}{4 ε_{2}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.30) & k \int_{Ω} (φ_{x} + ψ) (g_{1} ◇ ψ) d x ⩽ ε_{2} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + \frac{k^{2} C_{p}}{4 ε_{2}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ - \int_{Ω} (g_{1} * ψ_{x}) (g_{1} ◇ ψ_{x}) d x \\ ⩽ \frac{1}{2 ε_{2}} \int_{Ω} {(g_{1} ◇ ψ_{x})}^{2} d x + \frac{ε_{2}}{2} \int_{Ω} [\int_{0}^{t} g_{1} (t - s) {(ψ_{x} (t) - (ψ_{x} (t) - ψ_{x} (s)) d s]}^{2} d x \\ ⩽ ε_{2} \int_{0}^{t} g_{1} (s) d s \int_{Ω} ψ_{x}^{2} d x + (ε_{2} + \frac{1}{2 ε_{2}}) \int_{Ω} {(g_{1} ◇ ψ_{x})}^{2} d x \\ (4.31) & ⩽ c_{0} ε_{2} \int_{Ω} ψ_{x}^{2} d x + c_{0} (ε_{2} + \frac{1}{ε_{2}}) \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.32) & - \int_{Ω} g (g_{1} ◇ ψ) d x ⩽ C_{ε_{2}} \int_{Ω} g^{2} d x + ε_{2} \int_{Ω} (g_{1} \circ ψ_{x}) d x . \end{array}$ Substituting (4.27)–(4.32) into (4.26), we can obtain (4.25). □
Lemma 4.5.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then the function $\begin{matrix} (4.33) & I_{4} (t) = ρ_{3} \int_{Ω} θ_{t} θ d x + γ \int_{Ω} (φ_{x} + ψ) θ d x, t ⩾ 0, \end{matrix}$ we have the following estimate, for any $ε_{3} > 0$ , $\begin{matrix} (4.34) & I_{4}^{'} (t) ⩽ - \frac{λ_{2}}{2} \int_{Ω} θ_{x}^{2} d x + c_{0} (1 + \frac{1}{ε_{3}}) \int_{Ω} θ_{t}^{2} d x + ε_{3} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + c_{0} \int_{Ω} (g_{2} \circ θ_{x} + h^{2}) d x . \end{matrix}$
Proof.
Differentiating $I_{4} (t)$ and using (1.2), we can obtain $\begin{array}{l} I_{4}^{'} (t) = & ρ_{3} \int_{Ω} θ_{t}^{2} d x - (b_{2} - \int_{0}^{t} g_{2} (s) d s) \int_{Ω} θ_{x}^{2} d x + γ \int_{Ω} (φ_{x} + ψ) θ_{t} d x \\ (4.35) & + \int_{Ω} h θ d x - \int_{Ω} (g_{2} ◇ θ_{x}) θ_{x} d x . \end{array}$ Then using Young’s and Poincaré’s inequalities, Lemma 2.1, we can obtain for any $ε_{3}, η > 0$ , $\begin{array}{l} (4.36) & γ \int_{Ω} (φ_{x} + ψ) θ_{t} d x ⩽ \frac{γ^{2}}{4 ε_{3}} \int_{Ω} θ_{t}^{2} d x + ε_{3} \int_{Ω} {(φ_{x} + ψ)}^{2} d x, \\ (4.37) & - \int_{Ω} (g_{2} ◇ θ_{x}) θ_{x} d x ⩽ \frac{c_{0}}{4 η} \int_{Ω} (g_{2} \circ θ_{x}) d x + η \int_{Ω} θ_{x}^{2} d x, \\ (4.38) & \int_{Ω} h θ d x ⩽ \frac{C_{p}}{4 η} \int_{Ω} h^{2} d x + η \int_{Ω} θ_{x}^{2} d x . \end{array}$ Substituting (4.36)–(4.38) into (4.35) and choosing $η = \frac{λ_{2}}{4}$ , from $(H_{1})$ we can conclude (4.34). □
Lemma 4.6.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), then the function $\begin{matrix} (4.39) & I_{5} (t) = - ρ_{3} \int_{Ω} θ_{t} (g_{2} ◇ θ) d x, t ⩾ 0, \end{matrix}$ satisfies for any positive constants $ε_{3}$ , $δ_{2} > 0$ , $\begin{array}{l} I_{5}^{'} (t) ⩽ & - (ρ_{3} \int_{0}^{t} g_{2} (s) d s - δ_{2}) \int_{Ω} θ_{t}^{2} d x + c_{0} ε_{3} \int_{Ω} θ_{x}^{2} d x + ε_{3} \int_{Ω} φ_{t}^{2} d x + ε_{3} \int_{Ω} ψ_{t}^{2} d x \\ (4.40) & + c_{0} (ε_{3} + \frac{1}{ε_{3}}) \int_{Ω} (g_{2} \circ θ_{x}) d x - c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x + C_{ε_{3}} \int_{Ω} h^{2} d x . \end{array}$
Proof.
Differentiating $I_{5} (t)$ and using (1.2), we obtain $\begin{array}{l} I_{5}^{'} (t) = & - (ρ_{3} \int_{0}^{t} g_{2} (s) d s) \int_{Ω} θ_{t}^{2} d x - ρ_{3} \int_{Ω} θ_{t} (g_{2}^{'} ◇ θ) d x \\ + b_{2} \int_{Ω} θ_{x} (g_{2} ◇ θ_{x}) d x - γ \int_{Ω} φ_{t} (g_{2} ◇ θ_{x}) d x \\ (4.41) & + γ \int_{Ω} ψ_{t} (g_{2} ◇ θ) d x - \int_{Ω} (g_{2} * θ_{x}) (g_{2} ◇ θ_{x}) d x - \int_{Ω} h (g_{2} ◇ θ) d x . \end{array}$ Using Young’s inequality, Poincaré’s inequality, Lemma 2.1, and Remark 2.1, we can obtain $\begin{array}{l} (4.42) & - ρ_{3} \int_{Ω} θ_{t} (g_{2}^{'} ◇ θ) d x ⩽ δ_{2} \int_{Ω} θ_{t}^{2} d x - \frac{c_{0}}{δ_{2}} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x, \\ (4.43) & b_{2} \int_{Ω} θ_{x} (g_{2} ◇ θ_{x}) d x ⩽ ε_{3} \int_{Ω} θ_{x}^{2} d x + \frac{b_{2}^{2} c_{0}}{4 ε_{3}} \int_{Ω} (g_{2} \circ θ_{x}) d x, \\ (4.44) & - γ \int_{Ω} φ_{t} (g_{2} ◇ θ_{x}) d x ⩽ ε_{3} \int_{Ω} φ_{t}^{2} d x + \frac{γ^{2} c_{o}}{4 ε_{3}} \int_{Ω} (g_{2} \circ θ_{x}) d x, \\ (4.45) & γ \int_{Ω} ψ_{t} (g_{2} ◇ θ) d x ⩽ ε_{3} \int_{Ω} ψ_{t}^{2} d x + \frac{γ^{2} c_{0}}{4 ε_{3}} \int_{Ω} (g_{2} \circ θ_{x}) d x, \\ (4.46) & - \int_{Ω} h (g_{2} ◇ θ) d x ⩽ ε_{3} \int_{Ω} (g_{2} \circ θ_{x}) d x + C_{ε_{3}} \int_{Ω} h^{2} d x, \\ (4.47) & - \int_{Ω} (g_{2} * θ_{x}) (g_{2} ◇ θ_{x}) d x ⩽ c_{0} ε_{3} \int_{Ω} θ_{x}^{2} d x + c_{0} (ε_{3} + \frac{1}{ε_{3}}) \int_{Ω} (g_{2} \circ θ_{x}) d x . \end{array}$ Substituting (4.42)–(4.47) into (4.41), for any $δ_{2}$ , $ε_{3} > 0$ , then we can conclude (4.40). □
Lemma 4.7.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), assume that the coefficients satisfy ( 4.6 ), then the function $\begin{array}{l} I_{6} = & ρ_{2} \int_{Ω} ψ_{t} (φ_{x} + ψ) d x + (ρ_{2} - γ) \int_{Ω} φ_{t} ψ_{x} d x - (\frac{γ}{k} + \frac{ρ_{3}}{ρ_{2}}) \int_{Ω} θ_{x} (g_{1} * ψ_{x}) d x \\ (4.48) & - \frac{ρ_{1}}{k} \int_{Ω} φ_{t} (g_{1} * ψ_{x}) d x + ρ_{3} \int_{Ω} ψ_{t} θ_{t} d x + b_{2} \int_{Ω} ψ_{x} θ_{x} d x - \int_{Ω} ψ_{x} (g_{2} * θ_{x}) d x, \end{array}$ satisfies for any positive constant $ε_{4} > 0$ , $\begin{array}{l} I_{6}^{'} (t) & ⩽ - \frac{k}{2} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + ρ_{2} \int_{Ω} ψ_{t}^{2} d x + c_{0} \int_{Ω} θ_{t}^{2} d x + ε_{4} \int_{Ω} φ_{t}^{2} d x + ε_{4} \int_{Ω} θ_{x}^{2} d x \\ + c_{0} (1 + \frac{1}{ε_{4}}) \int_{Ω} ψ_{x}^{2} d x - \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x - c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x + \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1} \circ ψ_{x}) d x \\ (4.49) & + ε_{4} \int_{Ω} f^{2} d x + c_{0} \int_{Ω} (g^{2} + h^{2}) d x + [φ_{x} (b_{1} ψ_{x} - (g_{1} * ψ_{x}))] |_{x = 0}^{x = L} . \end{array}$
Proof.
By a direct computation and using (1.2)₁ and (1.2)₂, yield $\begin{array}{rcl} I_{6}^{'} (t) & = & [φ_{x} (b_{1} ψ_{x} - (g_{1} * ψ_{x}))] |_{x = 0}^{x = L} - k \int_{Ω} {(φ_{x} + ψ)}^{2} d x + (ρ_{2} - γ) \int_{Ω} ψ_{t}^{2} d x + \frac{ρ_{3} γ}{ρ_{2}} \int_{Ω} θ_{x}^{2} d x \\ - \underset{κ_{1}}{\underset{︸}{(γ + \frac{b_{1} ρ_{1}}{k} - ρ_{2})}} \int_{Ω} φ_{t t} ψ_{x} d x - \underset{κ_{2}}{\underset{︸}{(\frac{ρ_{3} b_{1}}{ρ_{2}} + \frac{b_{1} γ}{k} - b_{2})}} \int_{Ω} ψ_{x} θ_{t x} d x - \frac{μ b_{1}}{k} \int_{Ω} ψ_{x} φ_{t} d x \\ - \frac{ρ_{1}}{k} \int_{Ω} φ_{t} (g_{1} ψ_{x} - g_{1}^{'} ◇ ψ_{x}) d x - \int_{Ω} ψ_{x} (g_{2} θ_{x} - g_{2}^{'} ◇ θ_{x}) d x + \frac{μ}{k} \int_{Ω} φ_{t} (g_{1} * ψ_{x}) d x \\ + (γ - \frac{ρ_{3} k}{ρ_{2}}) \int_{Ω} θ_{t} (φ_{x} + ψ) d x - (\frac{γ}{k} + \frac{ρ_{3}}{ρ_{2}}) \int_{Ω} θ_{x} (g_{1} ψ_{x} - g_{1}^{'} ◇ ψ_{x}) d x \\ + \frac{b_{1}}{k} \int_{Ω} f ψ_{x} d x + \int_{Ω} g (φ_{x} + ψ) d x + \frac{ρ_{3}}{ρ_{2}} \int_{Ω} g θ_{t} d x \\ (4.50) & + \int_{Ω} h ψ_{t} d x - \frac{1}{k} \int_{Ω} f (g_{1} * ψ_{x}) d x . \end{array}$ Then using Young’s inequality and Lemma 2.2, we can obtain $\begin{matrix} (4.51) & (γ - \frac{ρ_{3} k}{ρ_{2}}) \int_{Ω} θ_{t} (φ_{x} + ψ) d x ⩽ c_{0} \int_{Ω} θ_{t}^{2} d x + \frac{k}{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x, \end{matrix}$ and for any constant $ε_{4} > 0$ , we have $\begin{array}{l} (4.52) & - \frac{ρ_{1}}{k} \int_{Ω} φ_{t} (g_{1} ψ_{x} - g_{1}^{'} ◇ ψ_{x}) d x ⩽ ε_{4} \int_{Ω} φ_{t}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x - \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x, \\ (4.53) & - \int_{Ω} ψ_{x} (g_{2} θ_{x} - g_{2}^{'} ◇ θ_{x}) d x ⩽ ε_{4} \int_{Ω} θ_{x}^{2} d x + c_{0} (1 + \frac{1}{ε_{4}}) \int_{Ω} ψ_{x}^{2} d x - c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x, \\ - (\frac{γ}{k} + \frac{ρ_{3}}{ρ_{2}}) \int_{Ω} θ_{x} (g_{1} ψ_{x} - g_{1}^{'} ◇ ψ_{x}) d x \\ (4.54) & ⩽ ε_{4} \int_{Ω} θ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x - \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x, \\ \frac{μ}{k} \int_{Ω} φ_{t} (g_{1} * ψ_{x}) d x ⩽ \frac{ε_{4}}{2} \int_{Ω} φ_{t}^{2} d x + \frac{c_{0}}{2 ε_{4}} \int_{Ω} {(g_{1} * ψ_{x})}^{2} d x \\ = \frac{ε_{4}}{2} \int_{Ω} φ_{t}^{2} d x + \frac{c_{0}}{2 ε_{4}} \int_{Ω} {((\int_{0}^{t} g_{1} (s) d s) ψ_{x} - g_{1} ◇ ψ_{x})}^{2} d x \\ (4.55) & ⩽ \frac{ε_{4}}{2} \int_{Ω} φ_{t}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.56) & - \frac{μ b_{1}}{k} \int_{Ω} ψ_{x} φ_{t} d x ⩽ ε_{4} \int_{Ω} φ_{t}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x, \\ (4.57) & \int_{Ω} g (φ_{x} + ψ) d x ⩽ \frac{k}{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + \frac{1}{k} \int_{Ω} g^{2} d x, \\ (4.58) & \frac{ρ_{3}}{ρ_{2}} \int_{Ω} g θ_{t} d x ⩽ \frac{ρ_{3}^{2}}{4 k ρ_{2}^{2}} \int_{Ω} θ_{t}^{2} d x + k \int_{Ω} g^{2} d x, \\ (4.59) & \int_{Ω} h ψ_{t} d x ⩽ γ \int_{Ω} ψ_{t}^{2} d x + \frac{1}{4 γ} \int_{Ω} h^{2} d x, \\ (4.60) & - \frac{1}{k} \int_{Ω} f (g_{1} * ψ_{x}) d x ⩽ ε_{4} \int_{Ω} f^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.61) & \frac{b_{1}}{k} \int_{Ω} f ψ_{x} d x ⩽ ε_{4} \int_{Ω} f^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x . \end{array}$ Substituting (4.51)–(4.61) to (4.50), and using the conditions $κ_{1}$ and $κ_{2}$ , we can obtain (4.49). □

In order to handle the boundary term in (4.49), we introduce the function $\begin{matrix} (4.62) & q (x) = 2 - \frac{4 x}{L}, x \in Ω, \end{matrix}$ then we have the following lemma.
Lemma 4.8.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), the function $J_{1}$ defined by $\begin{matrix} (4.63) & J_{1} = \frac{ε_{4}}{k} [ρ_{1} b_{2} \int_{Ω} φ_{t} q φ_{x} d x + γ \int_{Ω} φ_{x} q (b_{2} θ_{x} - g_{2} * θ_{x}) d x + ρ_{3} \int_{Ω} θ_{t} q (b_{2} θ_{x} - g_{2} * θ_{x}) d x], \end{matrix}$ satisfies for any $ε_{4} > 0$ , $\begin{array}{rcl} J_{1}^{'} (t) & ⩽ & - ε_{4} b_{2} [φ_{x}^{2} (L) + φ_{x}^{2} (0)] + \frac{1}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{t}^{2} d x \\ - ε_{4} c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x + C_{ε_{4}} \int_{Ω} f^{2} d x + C_{ε_{4}} \int_{Ω} h^{2} d x \\ (4.64) & + c_{0} ε_{4} {\int_{Ω} φ_{x}^{2} d x + \int_{Ω} φ_{t}^{2} d x + \int_{Ω} θ_{x}^{2} d x + \int_{Ω} θ_{t}^{2} d x + \int_{Ω} (g_{2} \circ θ_{x}) d x} . \end{array}$
Proof.
Taking the derivation of $J_{1} (t)$ and using (1.2)₁ and (1.2)₃, we obtain $\begin{array}{l} J_{1}^{'} (t) = & ε_{4} b_{2} \int_{Ω} q φ_{x} φ_{x x} d x + ε_{4} b_{2} \int_{Ω} q φ_{x} ψ_{x} d x - \frac{ε_{4} μ b_{2}}{k} \int_{Ω} q φ_{x} φ_{t} d x + \frac{ε_{4} b_{2}}{k} \int_{Ω} q f φ_{x} d x \\ + \frac{ε_{4} ρ_{1} b_{2}}{k} \int_{Ω} q φ_{t} φ_{x t} d x - \frac{ε_{4} γ}{k} \int_{Ω} q φ_{x} {(g_{2} * θ_{x})}^{'} d x + \frac{ε_{4}}{k} \int_{Ω} q h (b_{2} θ_{x} - (g_{2} * θ_{x})) d x \\ + \frac{ε_{4}}{k} \int_{Ω} q (b_{2} θ_{x} - (g_{2} * θ_{x})) (b_{2} θ_{x x} - g_{2} * θ_{x x}) d x - \frac{ε_{4} γ}{k} \int_{Ω} q ψ_{t} (b_{2} θ_{x} - (g_{2} * θ_{x})) d x \\ (4.65) & + \frac{ε_{4} ρ_{3} b_{2}}{k} \int_{Ω} q θ_{t} θ_{x t} d x - \frac{ε_{4} ρ_{3}}{k} \int_{Ω} q θ_{t} {(g_{2} * θ_{x})}^{'} d x . \end{array}$ Integrating by parts over Ω in (4.65), we can obtain, $\begin{array}{l} J_{1}^{'} (t) ⩽ & - ε_{4} b_{2} [φ_{x}^{2} (L) + φ_{x}^{2} (0)] + \frac{2 ε_{4} b_{2}}{L} \int_{Ω} φ_{x}^{2} d x + \frac{2 ε_{4} ρ_{1} b_{2}}{k L} \int_{Ω} φ_{t}^{2} d x + \frac{2 ε_{4} ρ_{3} b_{2}}{k L} \int_{Ω} θ_{t}^{2} d x \\ + \frac{2 ε_{4}}{k L} \int_{Ω} {(b_{2} θ_{x} - g_{2} * θ_{x})}^{2} d x - \frac{ε_{4} γ}{k} \int_{Ω} q φ_{x} {(g_{2} * θ_{x})}^{'} d x - \frac{ε_{4} ρ_{3}}{k} \int_{Ω} q θ_{t} {(g_{2} * θ_{x})}^{'} d x \\ + ε_{4} b_{2} \int_{Ω} q φ_{x} ψ_{x} d x - \frac{ε_{4} γ}{k} \int_{Ω} q ψ_{t} (b_{2} θ_{x} - (g_{2} * θ_{x})) d x - \frac{ε_{4} μ b_{2}}{k} \int_{Ω} q φ_{x} φ_{t} d x \\ (4.66) & + \frac{ε_{4} b_{2}}{k} \int_{Ω} q f φ_{x} d x + \frac{ε_{4}}{k} \int_{Ω} q h (b_{2} θ_{x} - (g_{2} * θ_{x})) d x . \end{array}$ Thanks to Young’s inequality, Lemma 2.1 and Lemma 2.2, we obtain the following estimates $\begin{array}{l} - \frac{ε_{4} γ}{k} \int_{Ω} q φ_{x} {(g_{2} * θ_{x})}^{'} d x & = - \frac{ε_{4} γ}{k} g_{2} (t) \int_{Ω} q φ_{x} θ_{x} d x + \frac{ε_{4} γ}{k} \int_{Ω} q φ_{x} (g_{2}^{'} ◇ θ_{x}) d x \\ (4.67) & ⩽ ε_{4} c_{0} \int_{Ω} φ_{x}^{2} d x + ε_{4} c_{0} \int_{Ω} θ_{x}^{2} d x - ε_{4} c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x . \end{array}$ Similarly, we obtain $\begin{array}{l} - \frac{ε_{4} ρ_{3}}{k} \int_{Ω} q θ_{x} {(g_{2} * θ_{x})}^{'} d x = - \frac{ε_{4} ρ_{3}}{k} g_{2} (t) \int_{Ω} q θ_{t} θ_{x} d x + \frac{ε_{4} ρ_{3}}{k} \int_{Ω} q θ_{t} (g_{2}^{'} ◇ θ_{x}) d x \\ (4.68) & ⩽ ε_{4} c_{0} \int_{Ω} θ_{t}^{2} d x + ε_{4} c_{0} \int_{Ω} θ_{x}^{2} d x - ε_{4} c_{0} \int_{Ω} (g_{2}^{'} \circ θ_{x}) d x, \\ (4.69) & ε_{4} b_{2} \int_{Ω} q φ_{x} ψ_{x} d x ⩽ c_{0} ε_{4} \int_{Ω} φ_{x}^{2} d x + \frac{1}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x, \\ - \frac{ε_{4} γ}{k} \int_{Ω} q ψ_{t} (b_{2} θ_{x} - (g_{2} * θ_{x})) d x \\ (4.70) & ⩽ \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{t}^{2} d x + ε_{4} c_{0} \int_{Ω} θ_{x}^{2} d x + ε_{4} c_{0} \int_{Ω} (g_{2} \circ θ_{x}) d x . \end{array}$ Combining (4.67)–(4.70) to (4.66), by using Lemma 2.2, Poincaré’s inequality and Young’s inequality, we can obtain (4.64). □
Lemma 4.9.
Let $(φ, ψ, θ)$ be the solution of system ( 1.2 )–( 1.4 ), the function $J_{2}$ defined by $\begin{matrix} (4.71) & J_{2} (t) = \frac{ρ_{2}}{4 ε_{4}} \int_{Ω} ψ_{t} q (b_{1} ψ_{x} - (g_{1} * ψ_{x})) d x, t ⩾ 0, \end{matrix}$ satisfies for any $ε_{4} > 0$ , $\begin{array}{l} J_{2}^{'} (t) ⩽ & - \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (L) - g_{1} * ψ_{x} {(L)]}^{2} - \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (0) - g_{1} * ψ_{x} {(0)]}^{2} + \frac{c_{0}}{ε_{4}} \int_{Ω} θ_{t}^{2} d x \\ + c_{0} (1 + \frac{1}{ε_{4}}) \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} (1 + \frac{1}{ε_{4}^{2}}) \int_{Ω} (g_{1} \circ ψ_{x}) d x + ε_{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x \\ (4.72) & + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{t}^{2} d x - \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x + C_{ε_{4}} \int_{Ω} g^{2} d x . \end{array}$
Proof.
Taking the derivation of $J_{2} (t)$ , from (1.2)₂ and integrating by parts over Ω, we get $\begin{array}{l} J_{2}^{'} (t) = & - \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (L) - g_{1} * ψ_{x} {(L)]}^{2} - \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (0) - g_{1} * ψ_{x} {(0)]}^{2} \\ + \frac{1}{2 ε_{4} L} \int_{Ω} {(b_{1} ψ_{x} - g_{1} * ψ_{x})}^{2} d x - \frac{k}{4 ε_{4}} \int_{Ω} (φ_{x} + ψ) q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x \\ + \frac{γ}{4 ε_{4}} \int_{Ω} θ_{t} q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x + \frac{ρ_{2} b_{1}}{4 ε_{4}} \int_{Ω} ψ_{t} q {(b_{1} ψ_{x} - g_{1} * ψ_{x})}^{'} d x \\ (4.73) & + \frac{1}{4 ε_{4}} \int_{Ω} g q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x . \end{array}$ Then using Young’s inequality, Lemma 2.1, Lemma 2.2, and the fact that $q^{2} (x) ⩽ 4$ , $\forall x \in Ω$ , $\begin{array}{l} - \frac{k}{4 ε_{4}} \int_{Ω} (φ_{x} + ψ) q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x \\ (4.74) & ⩽ ε_{4} \int_{Ω} {(φ_{x} + ψ)}^{2} d x + \frac{c_{0}}{ε_{4}^{3}} \int_{Ω} (ψ_{x}^{2} + g_{1} \circ ψ_{x}) d x, \\ (4.75) & \frac{γ}{4 ε_{4}} \int_{Ω} θ_{t} q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x ⩽ \frac{c_{0}}{ε_{4}} \int_{Ω} θ_{t}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1} \circ ψ_{x}) d x, \\ (4.76) & \frac{ρ_{2} b_{1}}{4 ε_{4}} \int_{Ω} ψ_{t} q {(b_{1} ψ_{x} - g_{1} * ψ_{x})}^{'} d x ⩽ \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{t}^{2} d x + \frac{c_{0}}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x - \frac{c_{0}}{ε_{4}} \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x, \\ (4.77) & \frac{1}{4 ε_{4}} \int_{Ω} g q (b_{1} ψ_{x} - g_{1} * ψ_{x}) d x ⩽ \frac{1}{ε_{4}} \int_{Ω} ψ_{x}^{2} d x + \frac{1}{ε_{4}} \int_{Ω} (g_{1} \circ ψ_{x}) d x + C_{ε_{4}} \int_{Ω} g^{2} d x . \end{array}$ Finally substituting (4.74)–(4.77) into (4.73), we can obtain (4.72). □

Now, some positive constants N, $N_{1}$ , $N_{2}$ , $N_{3}$ and $N_{4}$ will be chosen appropriately later. For any $t > 0$ , we define the Lyapunov function by $\begin{matrix} (4.78) & L (t) : = N E + N_{1} I_{1} (t) + \frac{1}{4} I_{2} (t) + N_{2} I_{3} (t) + N_{3} I_{4} (t) + N_{4} I_{5} (t) + I_{6} (t) + J_{1} (t) + J_{2} (t) . \end{matrix}$ Proof of Theorem 4.1.
Taking into account Lemmas 4.1–4.9, and the following relations $\begin{array}{l} \int_{Ω} φ_{x}^{2} d x ⩽ 2 \int_{Ω} {(φ_{x} + ψ)}^{2} d x + 2 C_{p} \int_{Ω} ψ_{x}^{2} d x, \\ [φ_{x} (b_{1} ψ_{x} - g_{1} * ψ_{x})] |_{x = 0}^{x = L} ⩽ \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (L) - g_{1} * ψ_{x} {(L)]}^{2} \\ + \frac{1}{4 ε_{4}} [b_{1} ψ_{x} (0) - g_{1} * ψ_{x} {(0)]}^{2} + ε_{4} [φ_{x}^{2} (L) + φ_{x}^{2} (0)], \end{array}$ we obtain, $\begin{array}{rcl} L^{'} (t) & ⩽ & - [\frac{λ_{1} N_{1}}{2} - N_{2} c_{0} ε_{2} - N_{1} C_{p} ε_{1} - c_{0} - c_{0} (1 + \frac{1}{ε_{4}}) - c_{0} C_{p} ε_{4}] \int_{Ω} ψ_{x}^{2} d x \\ - [N_{2} (ρ_{2} \int_{0}^{t} g_{1} (s) d s - δ_{1}) - N ε - N_{1} c (ε_{1}) - N_{4} ε_{3} - c_{0} (1 + \frac{1}{ε_{4}})] \int_{Ω} ψ_{t}^{2} d x \\ - [N_{4} (ρ_{3} \int_{0}^{t} g_{2} (s) d s - δ_{2}) - N ε - N_{1} c_{0} - \frac{γ^{2}}{4 k} - N_{2} ε_{2} \\ - N_{3} c_{0} (1 + \frac{1}{ε_{3}}) - c_{0} - c_{0} (ε_{4} + \frac{1}{ε_{4}})] \int_{Ω} θ_{t}^{2} d x \\ - [\frac{k}{16} - N_{1} ε_{1} - N_{2} ε_{2} - N_{3} ε_{3} - c_{0} ε_{4}] \int_{Ω} {(φ_{x} + ψ)}^{2} d x \\ - [\frac{ρ_{1}}{4} - N ε - N_{1} ε_{1} - N_{4} ε_{3} - c_{0} ε_{4}] \int_{Ω} φ_{t}^{2} d x \\ - [\frac{λ_{2} N_{3}}{2} - N_{4} c_{0} ε_{3} - c_{0} ε_{4}] \int_{Ω} θ_{x}^{2} d x \\ + [N_{1} c_{0} + c_{0} + N_{2} c_{0} (ε_{2} + \frac{1}{ε_{2}}) + \frac{c_{0}}{ε_{4}} (1 + \frac{1}{ε_{4}^{2}})] \int_{Ω} (g_{1} \circ ψ_{x}) d x \\ + [N_{3} c_{0} + N_{4} c_{0} (ε_{3} + \frac{1}{ε_{3}}) + c_{0} ε_{4}] \int_{Ω} (g_{2} \circ θ_{x}) d x \\ + [\frac{N}{2} - N_{2} c_{0} - \frac{c_{0}}{ε_{4}}] \int_{Ω} (g_{1}^{'} \circ ψ_{x}) d x \\ + [\frac{N}{2} - N_{4} c_{0} - c_{0} - c_{0} ε_{4}] \int_{Ω} (g_{2}^{'} \circ ψ_{x}) d x \\ + [N C_{ε} + N_{1} c_{0} + ε_{4} + C_{ε_{4}} + c_{0}] \int_{Ω} f^{2} d x \\ + [N C_{ε} + N_{1} c_{0} + N_{2} C_{ε_{2}} + C_{ε_{4}} + c_{0}] \int_{Ω} g^{2} d x \\ (4.79) & + [N C_{ε} + N_{3} c_{0} + N_{4} C_{ε_{3}} + C_{ε_{4}} + c_{0}] \int_{Ω} h^{2} d x . \end{array}$

Since $g_{1}$ and $g_{2}$ are continuous and positive, then for any $t ⩾ t_{0} > 0$ , we have $\begin{matrix} g_{0} = min {\int_{0}^{t_{0}} g_{1} (s) d s, \int_{0}^{t_{0}} g_{2} (s) d s} . \end{matrix}$

Now, if we select constants carefully, all the terms to the right-hand side of (4.79) become negative.

First, we take $δ_{1} = \frac{ρ_{2}}{8 N_{2}}$ , $δ_{2} = \frac{γ^{2}}{8 N_{4}}$ , and choose ε, $ε_{1}$ small enough $\begin{matrix} ε ⩽ \frac{ρ_{1}}{16 N}, ε_{1} ⩽ min {\frac{ρ_{1}}{12 N_{1}}, \frac{k}{64 N_{1}}} . \end{matrix}$ Second, we select $N_{1}$ and $N_{3}$ large enough so that $\begin{array}{l} \frac{λ_{1} N_{1}}{2} - N_{2} c_{0} ε_{2} - c_{0} - c_{0} (1 + \frac{1}{ε_{4}}) - c_{0} ε_{4} > 0, \\ \frac{λ_{2} N_{3}}{2} - N_{4} c_{0} ε_{3} - c_{0} ε_{4} > 0 . \end{array}$ Now, we choose $ε_{2}$ , $ε_{3}$ so small that $\begin{matrix} ε_{2} < \frac{k}{64 N_{2}}, ε_{3} ⩽ min {\frac{k}{64 N_{3}}, \frac{ρ_{1}}{24 N_{4}}} . \end{matrix}$ After that, we take $N_{2}$ and $N_{4}$ large enough such that $\begin{array}{l} N_{2} g_{0} - N ε - N_{1} c (ε_{1}) - N_{4} ε_{3} - c_{0} (1 + \frac{1}{ε_{4}}) > 0, \\ N_{4} g_{0} - N ε - N_{1} c_{0} - N_{2} ε_{2} - N_{3} c_{0} (1 + \frac{1}{ε_{3}}) - c_{0} - c_{0} (ε_{4} + \frac{1}{ε_{4}}) > 0 . \end{array}$ Finally, we choose $ε_{4}$ small enough, and N large enough so that $\begin{matrix} \frac{N}{2} - N_{2} c_{0} - \frac{c_{0}}{ε_{4}} > 0, \frac{N}{2} - N_{4} c_{0} - c_{0} - c_{0} ε_{4} > 0 . \end{matrix}$

Therefore, for positive constants $K_{0}$ , $K_{1}$ , C, (4.79) takes the form $\begin{matrix} (4.80) & L^{'} (t) ⩽ - K_{0} E (t) + K_{1} \int_{Ω} (g_{1} \circ ψ_{x} + g_{2} \circ θ_{x}) d x + C (‖ f ‖_{L^{2}}^{2} + ‖ g ‖_{L^{2}}^{2} + ‖ h ‖_{L^{2}}^{2}), \forall t ⩾ t_{0} . \end{matrix}$ Let $F^{2} (t) = ‖ f ‖_{L^{2}}^{2} + ‖ g ‖_{L^{2}}^{2} + ‖ h ‖_{L^{2}}^{2}$ . Then multiplying (4.80) by $ζ (t)$ and using $(H_{1})$ , $(H_{2})$ , we get $\begin{array}{l} ζ (t) L^{'} (t) & ⩽ - K_{0} ζ (t) E (t) + K_{1} ζ (t) \int_{Ω} (g_{1} \circ ψ_{x} + g_{2} \circ θ_{x}) d x + C ζ (t) F^{2} (t) \\ ⩽ - K_{0} ζ (t) E (t) - K_{2} \int_{Ω} (g_{1}^{'} \circ ψ_{x} + g_{2}^{'} \circ θ_{x}) d x + C ζ (t) F^{2} (t) \\ ⩽ - K_{0} ζ (t) E (t) - K_{2} E^{'} (t) + C ζ (t) F^{2} (t), \end{array}$ that is, $\begin{matrix} [ζ (t) L (t) + K_{2} E {(t)]}^{'} - ζ^{'} (t) L (t) ⩽ - K_{0} ζ (t) E (t) + C ζ (t) F^{2} (t) . \end{matrix}$ Using the fact that $ζ^{'} (t) ⩽ 0$ , $\forall t ⩾ t_{0}$ , we can obtain $\begin{matrix} [ζ (t) L (t) + K_{2} E {(t)]}^{'} ⩽ - K_{0} ζ (t) E (t) + C ζ (t) F^{2} (t) . \end{matrix}$ Noting again that $\begin{matrix} R (t) : = ζ (t) L (t) + K_{2} E (t) \sim E (t), \end{matrix}$ we get for positive constant α and $C^{'}$ , $\begin{matrix} (4.81) & R^{'} (t) ⩽ - α ζ (t) R (t) + C^{'} (‖ f ‖_{L^{2}}^{2} + ‖ g ‖_{L^{2}}^{2} + ‖ h ‖_{L^{2}}^{2}), \end{matrix}$ where $C^{'} > 0$ is a constant independent of initial data. Using Lemma 3.5 in [25] for (4.81), the proof is thus complete. □

5. Uniform attractors

In this section, we will establish the existence of uniform attractors for non-autonomous system (1.2)–(1.4). Setting $u = φ_{t}$ , $v = ψ_{t}$ , $ω = θ_{t}$ , $ℝ_{τ} = [τ, + \infty), τ ⩾ 0$ , we study the following system $\begin{matrix} (5.1) & \{\begin{matrix} φ_{t} - u = 0, \\ ρ_{1} φ_{t t} - k {(φ_{x} + ψ)}_{x} + γ θ_{t x} + μ φ_{t} = f (x, t), \\ ψ_{t} - v = 0, \\ ρ_{2} ψ_{t t} - b_{1} ψ_{x x} + k (φ_{x} + ψ) - γ θ_{t} + \int_{0}^{t} g_{1} (t - s) ψ_{x x} (x, s) d s = g (x, t), \\ θ_{t} - ω = 0, \\ ρ_{3} θ_{t t} - b_{2} θ_{x x} + \int_{0}^{t} g_{2} (t - s) θ_{x x} (x, s) d s + γ φ_{t x} + γ ψ_{t} = h (x, t), \end{matrix} \end{matrix}$ with the initial conditions and boundary conditions $\begin{matrix} (5.2) & \{\begin{matrix} (φ (x, 0), ψ (x, 0), θ (x, 0)) = (φ_{0} (x), ψ_{0} (x), θ_{0} (x)), & \forall x \in Ω, \\ (φ_{t} (x, 0), ψ_{t} (x, 0), θ_{t} (x, 0)) = (φ_{1} (x), ψ_{1} (x), θ_{1} (x)), & \forall x \in Ω, \\ φ (0, t) = φ (L, t) = ψ (0, t) = ψ (L, t) = θ (0, t) = θ (L, t) = 0, & \forall t \in ℝ_{τ} . \end{matrix} \end{matrix}$

Let $\begin{array}{l} (5.3) & F = (\begin{matrix} 0 \\ \frac{f}{ρ_{1}} \\ 0 \\ \frac{g}{ρ_{2}} \\ 0 \\ \frac{h}{ρ_{3}} \end{matrix}) \in Y = L^{2} (ℝ_{τ}, {(L^{2} (0, L))}^{6}), \\ (5.4) & H_{1} = (H_{0}^{1} (0, 1) \times L^{2} {(0, 1))}^{3} . \end{array}$

The energy function of problem (5.1) is given by $\begin{array}{l} E (t) = & \frac{1}{2} \int_{Ω} {φ_{t}^{2} + ψ_{t}^{2} + θ_{t}^{2} + {(φ_{x} + ψ)}^{2} + (b_{1} - \int_{0}^{t} g_{1} (s) d s) ψ_{x}^{2} + (b_{2} - \int_{0}^{t} g_{2} (s) d s) θ_{x}^{2} \\ + g_{1} \circ ψ_{x} + g_{2} \circ θ_{x}} d x . \end{array}$

For any $(φ_{τ}, φ_{1 τ}, ψ_{τ}, ψ_{1 τ}, θ_{τ}, θ_{1 τ}) \in H_{1}$ and any $F \in Y$ , we define for any $t ⩾ τ$ , $τ ⩾ 0$ , $\begin{array}{l} U_{F} (t, τ) : (φ_{τ}, φ_{1 τ}, ψ_{τ}, ψ_{1 τ}, θ_{τ}, θ_{1 τ}) \\ \in H_{1} \mapsto (φ (t), φ_{t} (t), ψ (t), ψ_{t} (t), θ (t), θ_{t} (t)) \\ = U_{F} (t, τ) (φ_{τ}, φ_{1 τ}, ψ_{τ}, ψ_{1 τ}, θ_{τ}), \end{array}$ where $(φ (t), φ_{t} (t), ψ (t), ψ_{t} (t), θ (t), θ_{t} (t))$ solves problem (5.1).

Our result concerns the uniform attractor in $H_{1}$ , we define the hull of $F_{0} \in Y$ as $Σ = H (F_{0}) = {[F_{0} (t + h) | h \in ℝ_{+}]}_{Y}$ , where ${[\cdot]}_{Y}$ denotes the closure in Banach space Y.

We note that $\begin{matrix} F_{0} \in Y \subseteq \hat{Y} = L_{loc}^{2} (ℝ_{+}, {(L^{2} (0, L))}^{6}), \end{matrix}$ where $F_{0}$ is a translation compact function in the weak topology of $\hat{E}$ , which means that $H (F_{0})$ is compact in $\hat{Y}$ , we consider the Banach space $L_{loc}^{p} (ℝ_{+}, Y_{1})$ of functions $σ (s), s \in ℝ_{+}$ with values in Banach space $Y_{1}$ that are locally p-power integrable in the Bochner means.

Particularly, for any interval $[t_{1}, t_{2}] \subseteq ℝ_{+}$ , $\int_{t_{1}}^{t_{2}} ‖ σ (s) ‖_{Y_{1}}^{p} d s < + \infty$ . Let $σ (s) \in L_{loc}^{p} (ℝ_{+}, Y_{1})$ , we consider the quantity $\begin{matrix} ϵ_{σ} (h) = sup_{t \in ℝ_{+}} \int_{t}^{t + h} {‖ σ (s) ‖}_{Y_{1}}^{p} d s . \end{matrix}$

Lemma 5.1.
Let $Σ = {[F_{0} (t + h) | h \in ℝ_{+}]}_{Y}$ , where $F_{0} \in Y$ is an arbitrary but fixed symbol function. Then for any $F \in Σ$ and for any $(φ_{τ}, φ_{1 τ}, ψ_{τ}, ψ_{1 τ}, θ_{τ}, θ_{1 τ}) \in H_{1}, τ ⩾ 0$ . ( 5.1 ) admits a unique global solution $(φ (t), φ_{t} (t), ψ (t), ψ_{t} (t), θ (t), θ_{t} (t)) \in H_{1}$ , which generates a unique semi-processes $U_{F} (t, τ)$ ( $t ⩾ τ$ , $τ ⩾ 0$ ) on $H_{1}$ of a two-parameter family of operators, such that for any $t ⩾ τ$ , $τ ⩾ 0$ , $\begin{matrix} U_{F} (t, τ) (φ_{τ}, φ_{1 τ}, ψ_{τ}, ψ_{1 τ}, θ_{τ}, θ_{1 τ}) = (φ (t), φ_{t} (t), ψ (t), ψ_{t} (t), θ (t), θ_{t} (t)) \in H_{1}, \end{matrix}$ i.e., $\begin{array}{l} φ (t) \in C (ℝ_{τ}, H_{0}^{1} (0, 1)), φ_{t} (t) \in C (ℝ_{τ}, L^{2} (0, 1)), ψ (t) \in C (ℝ_{τ}, H_{0}^{1} (0, 1)), \\ ψ_{t} (t) \in C (ℝ_{τ}, L^{2} (0, 1)), θ (t) \in C (ℝ_{τ}, H_{0}^{1} (0, 1)), θ_{t} (t) \in C (ℝ_{τ}, L^{2} (0, 1)) . \end{array}$

Let $X$ be a Banach space and $\hat{Σ}$ be a parameter set. The operators $U_{F} (t, τ)$ ( $t ⩾ τ$ , $τ ⩾ 0$ , $F \in \hat{Σ}$ ) are said to be a family of processes in $X$ with symbol space $\hat{Σ}$ if for any $F \in \hat{Σ}$ , $\begin{array}{l} (5.5) & U_{F} (t, s) U_{F} (s, τ) = U_{F} (t, τ), \forall t ⩾ s ⩾ τ, τ ⩾ 0, \\ (5.6) & U_{F} (τ, τ) = Id (identity operator), \forall τ ⩾ 0 . \end{array}$
Lemma 5.2 (Chepyzhov and Vishik [4]).

Let Σ be defined as before and $F_{0} \in Y$ , then

$F_{0}$ is translation compact in $\hat{Y}$ and any $F \in Σ = H (F_{0})$ is also a translation compact function in $\hat{Y}$ , moreover, $H (F) \subseteq H (F_{0})$ ;

The set $H (F_{0})$ is bounded in $L^{2} (ℝ_{+}, {(L^{2} (0, 1))}^{6})$ such that $ϵ_{F} (h) ⩽ ϵ_{F_{0}} (h) < + \infty$ , $\forall F \in Σ$ .

Lemma 5.3 (Chueshov and Lasiecka [5]).

Let $U_{F} (t, τ)$ ( $F \in \hat{Σ}$ , $t ⩾ τ$ , $τ ⩾ 0$ ) be a family of processes satisfying the translation identities ( 5.5 ) and ( 5.6 ) on Banach space $X$ , and have a bounded uniformly (with respect to $F \in \hat{Σ}$ ) absorbing set $A_{0} \subseteq X$ . Moreover, assuming that for any $ε > 0$ , there exists a time $T = T (A_{0}, ε) > 0$ and a contractive function $ϕ_{T}$ on $A_{0} \times A_{0}$ such that $\begin{matrix} ‖ U_{F_{1}} (T, 0) x - U_{F_{2}} (T, 0) y ‖ ⩽ ε + ϕ_{T} (x, y; F_{1}, F_{2}), \forall x, y \in A_{0}, \forall F_{1}, F_{2} \in \hat{Σ} . \end{matrix}$ Then $U_{F} (t, τ)$ ( $F \in \hat{Σ}$ , $t ⩾ τ$ , $τ ⩾ 0$ ) is uniformly (w.r.t. $F \in \hat{Σ}$ ) asymptotically compact in $X$ .

Lemma 5.4 (Chepyzhov, Pata and Vishik [3]).

For every $τ \in ℝ$ , assume that $ϕ_{0}$ is non-negative locally summable function on $R_{τ} \equiv [τ, + \infty)$ , then for every $υ > 0$ and every $t ⩾ τ$ , we have $\begin{matrix} sup_{t ⩾ τ} \int_{τ}^{t} ϕ_{0} (s) e^{- υ (t - s)} d s ⩽ \frac{1}{1 - e^{- υ}} sup_{t ⩾ τ} \int_{t}^{t + 1} ϕ_{0} (s) d s . \end{matrix}$

Now, we shall establish that the family of semi-processes $U_{F} (t, τ)$ has a bounded uniformly absorbing set given in the following lemma.

Lemma 5.5.

Under the hypothesis ( 5.3 ), the process family $U_{F} (t, τ)$ ( $F \in Σ$ , $t ⩾ τ$ , $τ ⩾ 0$ ) corresponding to ( 5.1 )–( 5.2 ) has a bounded uniformly absorbing set $A_{0} \in H_{1}$ .

Proof.

Similarly to [25], we can complete the proof by using Lemmas 5.2–5.4. □

Lemma 5.6.

Assume that F satisfies ( 5.3 ), then the family of semi-process $U_{F} (t, τ)$ ( $F \in Σ$ , $t ⩾ τ$ , $τ ⩾ 0$ ), corresponding to ( 5.1 ), is uniformly (w.r.t. $F \in Σ$ ) asymptotically compact in $H_{1}$ .

Proof.

Without loss of generality, we deal with the strong solutions in the following part, while for weak solutions, it follows easily from a similar argument.

For any $(φ_{τ}^{i}, φ_{1 τ}^{i}, ψ_{τ}^{i}, ψ_{1 τ}^{i}, θ_{τ}^{i}, θ_{1 τ}^{i}) \in A_{0}$ , let $(φ_{i} (t), φ_{i τ} (t), ψ_{i} (t), ψ_{i τ} (t), θ_{i} (t), θ_{i τ} (t))$ be the corresponding solution to $F_{i} \in Σ$ with respect to initial data $(φ_{τ}^{i}, φ_{1 τ}^{i}, ψ_{τ}^{i}, ψ_{1 τ}^{i}, θ_{τ}^{i}, θ_{1 τ}^{i})$ , $i = 1, 2$ .

Let $Q (t) = {(π (t), λ (t), α (t))}^{T} = U_{1} (t) - U_{2} (t) = {(π_{1} (t) - π_{2} (t), λ_{1} (t) - λ_{2} (t), α_{1} (t) - α_{2} (t))}^{T}$ . For any $x \in (0, L)$ , $t \in (τ, + \infty)$ , $Q (t)$ satisfies $\begin{array}{l} (5.7) & \{\begin{matrix} ρ_{1} π_{t t} - k {(π_{x} + λ)}_{x} + γ α_{t x} + μ π_{t} = f_{1} (x, t) - f_{2} (x, t), \\ ρ_{2} λ_{t t} - b_{1} λ_{x x} + k (π_{x} + λ) - γ α_{t} + \int_{0}^{t} g_{1} (t - s) λ_{x x} (x, s) d s = g_{1} (x, t) - g_{2} (x, t), \\ ρ_{3} α_{t t} - b_{2} α_{x x} + \int_{0}^{t} g_{2} (t - s) α_{x x} (x, s) d s + γ π_{t x} + γ λ_{t} = h (x, t), \\ (π (x, τ), λ (x, τ), α (x, τ)) = (φ_{τ}^{1} (x) - φ_{τ}^{2} (x), ψ_{τ}^{1} (x) - ψ_{τ}^{2} (x), θ_{τ}^{1} (x) - θ_{τ}^{2} (x)), \\ (π_{t} (x, τ), λ_{t} (x, τ), α_{t} (x, τ)) = (φ_{1 τ}^{1} (x) - φ_{1 τ}^{2} (x), ψ_{1 τ}^{1} (x) - ψ_{1 τ}^{2} (x), θ_{1 τ}^{1} (x) - θ_{1 τ}^{2} (x)), \\ π (0, t) = π (L, t) = λ (0, t) = λ (L, t) = α (0, t) = α (L, t) = 0, \end{matrix} \\ E_{Q} (t) = \frac{1}{2} \int_{Ω} {π_{t}^{2} + λ_{t}^{2} + α_{t}^{2} + {(π_{x} + λ)}^{2} + (1 - \int_{0}^{t} g_{1} (s) d s) λ_{x}^{2} \\ + (1 - \int_{0}^{t} g_{2} (s) d s) α_{x}^{2} + g_{1} \circ λ_{x} + g_{2} \circ α_{x}} d x . \end{array}$ We assume that $ρ_{1} = ρ_{2} = ρ_{3} = k = b_{1} = b_{2} = γ = 1$ , then similarly to Lemma 4.1, we have $\begin{array}{l} \frac{d E_{Q} (t)}{d t} = & - μ \int_{0}^{L} π_{t}^{2} d x - \frac{1}{2} g_{1} (t) \int_{0}^{L} λ_{x}^{2} d x - \frac{1}{2} g_{2} (t) \int_{0}^{L} α_{x}^{2} d x \\ + \frac{1}{2} \int_{0}^{L} (g_{1}^{'} \circ λ_{x}) d x + \frac{1}{2} \int_{0}^{L} (g_{2}^{'} \circ α_{x}) d x + \int_{0}^{L} (f_{1} - f_{2}) π_{t} d x \\ (5.8) & + \int_{0}^{L} (g_{1} - g_{2}) λ_{t} d x + \int_{0}^{L} (h_{1} - h_{2}) α_{t} d x . \end{array}$ Integrating (5.8) over $[σ, T]$ , where $0 ⩽ σ ⩽ T$ , we arrive at $\begin{array}{l} E_{Q} (T) + μ \int_{σ}^{T} \int_{0}^{L} π_{t}^{2} d x d t + \frac{1}{2} \int_{σ}^{T} g_{1} (t) \int_{0}^{L} λ_{x}^{2} d x d t + \frac{1}{2} \int_{σ}^{T} g_{2} (t) \int_{0}^{L} α_{x}^{2} d x d t \\ - \frac{1}{2} \int_{σ}^{T} \int_{0}^{L} (g_{1}^{'} \circ λ_{x}) d x d t - \frac{1}{2} \int_{σ}^{T} \int_{0}^{L} (g_{2}^{'} \circ α_{x}) d x d t \\ = \int_{σ}^{T} \int_{0}^{L} (f_{1} - f_{2}) π_{t} d x d t + \int_{σ}^{T} \int_{0}^{L} (g_{1} - g_{2}) λ_{t} d x d t \\ (5.9) & + \int_{σ}^{T} \int_{0}^{L} (h_{1} - h_{2}) α_{t} d x d t + E_{Q} (σ) . \end{array}$ Integrating (5.9) over $[0, T]$ with respect to σ, using Young’s inequality and $(H_{2})$ , we obtain $\begin{array}{rcl} T E_{Q} (t) & ⩽ & \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} (f_{1} - f_{2}) π_{t} d x d t d σ + \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} (g_{1} - g_{2}) λ_{t} d x d t d σ \\ + \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} (h_{1} - h_{2}) α_{t} d x d t d σ + \int_{0}^{T} E_{Q} (σ) d σ \\ ⩽ & \int_{0}^{T} E_{Q} (σ) d σ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(f_{1} - f_{2})}^{2} d x d t d σ \\ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(g_{1} - g_{2})}^{2} d x d t d σ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(h_{1} - h_{2})}^{2} d x d t d σ \\ (5.10) & + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} (π_{t}^{2} + λ_{t}^{2} + α_{t}^{2}) d x d t d σ . \end{array}$ Then we can obtain $\begin{array}{rcl} T E_{Q} (t) & ⩽ & \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(f_{1} - f_{2})}^{2} d x d t d σ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(g_{1} - g_{2})}^{2} d x d t d σ \\ (5.11) & + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(h_{1} - h_{2})}^{2} d x d t d σ + \int_{0}^{T} \int_{σ}^{T} E_{Q} (t) d t d σ + \int_{0}^{T} E_{Q} (σ) d σ . \end{array}$ From (5.7)–(5.8), it follows that $\begin{array}{l} \int_{0}^{T} | E_{Q} (σ) | d σ & = \int_{0}^{T} E_{Q} (t) d σ ⩽ \int_{0}^{T} (E_{U_{1}} (σ) + E_{U_{2}} (σ)) d σ \\ ⩽ 2 \int_{0}^{T} [E (τ) e^{- γ (t - τ)} + C \int_{τ}^{σ} (‖ f ‖_{2}^{2} + ‖ g ‖_{2}^{2} + ‖ h ‖_{2}^{2}) e^{- γ (t - s)} d σ \\ (5.12) & ⩽ C_{1}, \end{array}$ where $C_{1} = C (T, τ, γ)$ is a positive constant.

Similarly to (5.12), we can obtain $\begin{array}{l} \int_{0}^{T} \int_{σ}^{T} E_{Q} (t) d t d σ & ⩽ 2 \int_{0}^{T} \int_{σ}^{T} E (τ) e^{- γ (t - τ)} d t d σ \\ + 2 C \int_{0}^{T} \int_{σ}^{T} \int_{τ}^{t} (‖ f ‖_{2}^{2} + ‖ g ‖_{2}^{2} + ‖ h ‖_{2}^{2}) e^{- γ (t - s)} d s d t d σ \\ (5.13) & ⩽ C_{2} . \end{array}$ Thus we conclude $\begin{array}{l} T E_{Q} (t) ⩽ & \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(f_{1} - f_{2})}^{2} d x d t d σ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(g_{1} - g_{2})}^{2} d x d t d σ \\ (5.14) & + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(h_{1} - h_{2})}^{2} d x d t d σ + C_{M} . \end{array}$ Let $\begin{array}{l} R ((φ_{0}^{1}, φ_{10}^{1}, ψ_{0}^{1}, ψ_{10}^{1}, θ_{0}^{1}, θ_{10}^{1}), (φ_{0}^{2}, φ_{10}^{2}, ψ_{0}^{2}, ψ_{10}^{2}, θ_{0}^{2}, θ_{10}^{2}), F_{1}, F_{2}) \\ = \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(f_{1} - f_{2})}^{2} d x d t d σ + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(g_{1} - g_{2})}^{2} d x d t d σ \\ (5.15) & + \frac{1}{2} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(h_{1} - h_{2})}^{2} d x d t d σ . \end{array}$ Thus we get $\begin{matrix} (5.16) & E_{Q} (T) ⩽ \frac{C_{M}}{T} + \frac{1}{T} R ((φ_{0}^{1}, φ_{10}^{1}, ψ_{0}^{1}, ψ_{10}^{1}, θ_{0}^{1}, θ_{10}^{1}), (φ_{0}^{2}, φ_{10}^{2}, ψ_{0}^{2}, ψ_{10}^{2}, θ_{0}^{2}, θ_{10}^{2}), F_{1}, F_{2}) . \end{matrix}$

Since the family of semi-process $U_{F} (t, τ)$ ( $F \in Σ$ , $t ⩾ τ$ , $τ ⩾ 0$ ) has a bounded uniformly absorbing set, by the definition of $C_{M}$ , we know that for any fixed $ε > 0$ , we can choose $T > 0$ so large that $\frac{C_{M}}{T} ⩽ ε$ . Then we can prove that $R (\cdot, \cdot, \cdot, \cdot, \cdot, \cdot) \in Contr (A_{0}, Σ)$ for every fixed T by Lemma 5.2. From the proof of Lemma 5.1, we can deduce that for any fixed T, we have $\begin{matrix} (5.17) & ⋃_{F \in Σ} ⋃_{t \in [τ, T]} U_{F} (t, τ) A_{0} \end{matrix}$ is bounded in $H_{1}$ and the bound depends on T.

Let $(φ_{i}, φ_{i t}, ψ_{i}, ψ_{i t}, θ_{i}, θ_{i t})$ be the solutions corresponding to initial data $(φ_{τ}^{i}, φ_{1 τ}^{i}, ψ_{τ}^{i}, ψ_{1 τ}^{i}, θ_{τ}^{i}, θ_{1 τ}^{i}) \in A_{0}$ with respect to symbol $F_{n} \in Σ$ , $n = 1, 2, \dots$ . Then from the existence theory of global solutions, we can derive $\begin{array}{l} (5.18) & lim_{i \to \infty} lim_{j \to \infty} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(f_{i} - f_{j})}^{2} d x d t d σ = 0, \\ (5.19) & lim_{i \to \infty} lim_{j \to \infty} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(g_{i} - g_{j})}^{2} d x d t d σ = 0, \\ (5.20) & lim_{i \to \infty} lim_{j \to \infty} \int_{0}^{T} \int_{σ}^{T} \int_{0}^{L} {(h_{i} - h_{j})}^{2} d x d t d σ = 0 . \end{array}$ By Theorem 4.2 in [28], the proof is now complete. □

Theorem 5.1.

Assume that f, g, h satisfies ( 5.3 ) and Σ is defined above, then the family of processes $U_{F} (t, τ)$ ( $F \in Σ$ , $t ⩾ τ$ , $τ ⩾ 0$ ) corresponding to ( 5.1 ) has a compact uniform (w.r.t. $F \in Σ$ ) attractor $A_{Σ}$ .

Proof.

Lemmas 5.5 and 5.6 imply the existence of a uniform attractor immediately. □

Footnotes

Acknowledgements

This paper was supported in part by the NNSF of China with contract number 11671075 and the Fundamental Research Funds for the Central Universities.

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Global existence,asymptotic behavior and uniform attractor for a non-autonomous Timoshenko system of type III with weak damping

Abstract

Keywords

1. Introduction

2. Preliminaries

Lemma 2.1 (Fatori and Muñoz Rivera [6]).

Remark 2.1. Using Lemma 2.1 and Poincaré’s inequality, with − g ′ instead of g, we can obtain ∫ Ω ( ∫ 0 t − g ′ ( t − s ) ( h ( t ) − h ( s ) ) d s ) 2 d x ⩽ c 0 ∫ Ω ( g ′ ∘ h x ) d x . Lemma 2.2 (Ghennam and Djebabla [8]).

3. Well-posedness

Lemma 3.2 (Zheng [30]).

Proof of Theorem 3.1. According to Lemma 3.1 (see also [7]), we can know that A is an maximal monotone operator. By the assumptions, we have ( φ 0 , φ 1 , ψ 0 , ψ 1 , θ 0 , θ 1 ) T ∈ D ( A ) , and then by using Lemma 3.2, we complete the proof. □ 4. Uniform stability

Lemma 5.3 (Chueshov and Lasiecka [5]).

Lemma 5.4 (Chepyzhov, Pata and Vishik [3]).

Footnotes

Acknowledgements

References

Remark 2.1.
Using Lemma 2.1 and Poincaré’s inequality, with $- g^{'}$ instead of g, we can obtain $\begin{matrix} \int_{Ω} (\int_{0}^{t} - g^{'} (t - s) (h (t) - h {(s)) d s)}^{2} d x ⩽ c_{0} \int_{Ω} (g^{'} \circ h_{x}) d x . \end{matrix}$
Lemma 2.2 (Ghennam and Djebabla [8]).

Proof of Theorem 3.1.
According to Lemma 3.1 (see also [7]), we can know that $A$ is an maximal monotone operator. By the assumptions, we have ${(φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, θ_{0}, θ_{1})}^{T} \in D (A)$ , and then by using Lemma 3.2, we complete the proof. □

4. Uniform stability