Global attractors for a partially damped Timoshenko–Ehrenfest system without the hypothesis of equal wave speeds

Abstract

This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.

Keywords

Timoshenko-type systems gradient systems well-posedness quasi-stability global attractors

1. Introduction

In the literature it is shown that in general the linear Timoshenko system [32] introduced his classical equations which take into account both shear deformation and rotary inertia. Timoshenko had two predecessors [17,18], namely, Bresse [12] and Rayleigh [28] but he did not reference Bresse, though he sometimes referenced Rayleigh. On the other hand, Ehrenfest’s name did not appear in his papers dated 1920 and 1921. Koiter [21] did not know these facts when he wrote: “What is generally known as Timoshenko beam theory is a good example of a basic principle in the history of science: a theory which bears someone’s name is most likely due to someone else”. Elishakoff [16] unequivocally proves that the modern theory with the shear coefficient was introduced by Timoshenko and Ehrenfest. It is therefore fair that the theory should be called the Timoshenko–Ehrenfest theory is not exponentially stable, that is, the energy does not decay at an exponential rate for the cases of the so called partially dissipative Timoshenko system. To overcome this loss of stability, a hypothesis of equal wave speeds is necessary $\begin{matrix} (1) & \begin{matrix} v_{1}^{2} = v_{2}^{2}, \end{matrix} \end{matrix}$ where $v_{1}^{2} : = \frac{ρ_{1}}{κ}$ and $v_{2}^{2} : = \frac{ρ_{2}}{b}$ are the wave propagation speeds for stress waves and shear waves, respectively, of the classical beam model $\begin{array}{l} (2) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0 in (0, L) \times (0, \infty), \\ (3) & ρ_{2} ψ_{t t} - b ψ_{x x} + κ (φ_{x} + ψ) = 0 in (0, L) \times (0, \infty), \end{array}$ where ${(\cdot)}_{x}$ and ${(\cdot)}_{t}$ denote the derivative with respect to space and time, respectively. Moreover, $ρ_{1} = ρ A$ , $b = E I$ , $ρ_{2} = ρ I$ where A is the cross-sectional area, ρ is the density of the material, E is the Young’s modulus, I is the geometric moment of inertia, φ stands for the vertical displacement and ψ for the angular rotation.

Since the Soufyane’s paper [31], several researchers have been made notable contributions for the analysis of dissipative versions of the classical Eqs (2)–(3). In the past 20 years the studies distinguished primarily by the nature of the coupling between equations and the type or strength of damping. In a particular scenario, under the premise that Eqs (2)–(3) constitute a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the stabilization properties of the so-called partially damped Timoshenko systems (see [8–10,23,24,29] to quote a few) where some damping mechanism works on angle rotation or vertical displacement. In these cases, the exponential decay property of the energy solutions is achieved when the non-physical equal wave speeds assumption (1) plays the role to stabilization. Therefore, the most of the results constitute only a speculation sound because in practice the wave speeds are not equal. Of course for full damping cases the assumption (1) is not required [27].

Recent advances due to Almeida Júnior et al. [1–7] show new research directions for analysis of stabilization for Timoshenko–Ehrenfest type systems mainly based on damaging consequences of the so called second spectrum of frequency. This second mode of vibration or simply second spectrum appears as a lost element in the stabilization analysis and now it is clear why damping mechanisms are capable of producing exponential decay of energy in the partially dissipative cases. The main conclusions made by Almeida Júnior and Ramos [4] are:

The classical Timoshenko beam model predicts two phase velocities which governs the wave propagation in beams. The second phase velocity goes to infinity for lower wave numbers because the respective frequency has a non-null value for wave number equal to zero.

The damping effect acting on angle rotation of the classical Timoshenko beam model is able of eliminating the blow-up on second phase velocity for lower wave numbers. The same result holds for feedback law acting on vertical displacement [6].

The truncated version for Timoshenko model [15] given by $\begin{array}{l} (4) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (5) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) = 0, \end{array}$ is more simpler than classical ones because it has only one phase velocity. Therefore, the equal wave speed assumption is not required for getting the exponential decay for energy solutions.

The same above results hold for the Shear beam model according paper due to Almeida Júnior et al. [7] $\begin{array}{l} (6) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (7) & - b ψ_{x x} + κ (φ_{x} + ψ) = 0 . \end{array}$

Based on above explanations, for dissipative systems associated to Eqs (4)–(5) or to Eqs (6)–(7) it is expected that the exponential decay for energy solutions holds regardless any relationship between coefficients. In particular, Almeida Júnior et al. [7] showed that the Ostrogradsky’s energy [25] is negative when evaluated with the second spectrum concerning of the Eqs (2)–(3) and is positive when evaluated with the only spectrum concerning to the Eqs (4)–(5). For a deep discussion about the second spectrum of frequency we indicate the paper due to Bhaskar [11] as well as reference therein. Regarding studies about truncated versions for Timoshenko beam theory we indicate the paper due to Elishakoff and collaborators [15].

This paper is dedicated to study the existence of global attractors for nonlinear dynamics of the Timoshenko system $\begin{array}{l} (8) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0 in (0, L) \times (0, \infty), \\ (9) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) + μ ψ_{t} + f (ψ) = 0 in (0, L) \times (0, \infty), \end{array}$ where μ is the positive damping parameter and the nonlinear term $f (ψ)$ is the source term. Here, we assume the initial conditions $\begin{matrix} (10) & \begin{matrix} φ (0, x) = φ_{0} (x), φ_{t} (0, x) = φ_{1} (x), φ_{t t} (0, x) = φ_{2} (x), x \in (0, L), \\ ψ (0, x) = ψ_{0} (x), ψ_{t} (0, x) = ψ_{1} (x), x \in (0, L) \end{matrix} \end{matrix}$ and boundary conditions of Dirichlet–Neumann $\begin{array}{rcl} (11) & φ (0, t) = φ (L, t) = ψ_{x} (0, t) = ψ_{x} (L, t) = 0, t ⩾ 0 . \end{array}$

Note that the existence of global attractors for the classical Timoshenko system were recently studied by Fatori and Jorge Silva [19] in a fully damped setting, that is, with dissipation terms in all of its two equations. For the existence of global attractors of classical partially damped Timoshenko system, all literature results are devoted assuming the equal wave speed (1) see [20,33]. So, a natural question is whether an analogous result can be proven for the partially damped Timoshenko system (8)–(11) without assuming the equal wave speed (1). This was not considered before, and as we shall see, the mathematical analysis needed to establish the existence of global attractors is much more difficult.

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

All existence results in literature of global attractors for partially damped Timoshenko systems are devoted assuming the equal wave speed (1) see [20,33]. Here we obtain these same results without assuming (1).

In Theorem 2.2 we stablished the well-posedness of the problem (8)–(11). We note for this problem is not so trivial to get the Cauchy problem for the solution $(φ, φ_{t}, φ_{t t}, ψ)$ . Based on this limitation, because we can’t to apply the semigroup approach (see e.g., Pazy [26]), we use the classical Faedo–Galerkin method combined with compactness arguments.

We construct a smooth finite dimensional attractor for the system (8)–(11) using the recent quasi-stability theory [13,14].

2. Preliminaries and well-posedness

We use throughout this paper the standard Lebesgue spaces $L^{r} (0, L)$ , $r ⩾ 1$ , with the norm denoted by $‖ \cdot ‖_{r}$ . We denote by $(\cdot, \cdot)$ the inner product in $L^{2} (0, L)$ . Let us consider the spaces $\begin{array}{rcl} L_{*}^{2} (0, L) : = {u \in L^{2} (0, L); \int_{0}^{L} u (x) d x = 0}, \end{array}$ and $\begin{matrix} (12) & H_{*}^{1} (0, L) : = H^{1} (0, L) \cap L_{*}^{2} (0, L), H_{*}^{2} (0, L) : = H^{2} (0, L) \cap H_{*}^{1} (0, L) . \end{matrix}$ For the Sobolev space $H_{*}^{1} (0, L)$ the Poincaré’s inequality holds $\begin{matrix} λ_{1} ‖ u ‖_{2}^{2} ⩽ ‖ u_{x} ‖_{2}^{2}, \forall u \in H_{*}^{1} (0, L), \end{matrix}$ where $λ_{1} > 0$ is the Poincaré’s constant (the smallest eigenvalue of $- \partial_{x x}$ ). Therefore, $‖ u ‖_{H_{*}^{1} (0, L)} : = ‖ u_{x} ‖_{2}$ defines a norm in $H_{*}^{1} (0, L)$ .

Define the phase spaces $\begin{matrix} H = H_{0}^{1} (0, L) \times H_{0}^{1} (0, L) \times L^{2} (0, L) \times H_{*}^{1} (0, L), \end{matrix}$ and $\begin{matrix} H_{1} = {(H^{2} (0, L) \cap H_{0}^{1} (0, L))}^{2} \times H_{0}^{1} (0, L) \times H_{*}^{2} (0, L) . \end{matrix}$ The norm in $H$ is defined by $\begin{matrix} (13) & ‖ z ‖_{H}^{2} = ρ_{1} {‖ φ^{'} ‖}_{2}^{2} + \frac{ρ_{1} ρ_{2}}{2 κ} {‖ φ^{″} ‖}_{2}^{2} + b ‖ ψ_{x} ‖_{2}^{2} + κ ‖ φ_{x} + ψ ‖_{2}^{2}, \end{matrix}$ for all $z = (φ, φ^{'}, φ^{″}, ψ) \in H$ .

We assume that $f \in C^{1} (0, L)$ and there exist $θ ⩾ 1$ and $k_{0} > 0$ such that $\begin{matrix} (14) & | f^{'} (ψ) | ⩽ k_{0} (1 + | ψ |^{θ - 1}), \forall ψ \in R . \end{matrix}$ Moreover, letting $F (s) = \int_{0}^{s} f (r) d r$ , we assume that there exists constant $k_{1} > 0$ such that $\begin{matrix} (15) & \begin{matrix} F (ψ) ⩾ - k_{1} and f (ψ) ψ - F (ψ) ⩾ - k_{1}, \forall ψ \in R . \end{matrix} \end{matrix}$

2.1. Energy identities

Define the total energy of solutions $z = (φ, φ_{t}, φ_{t t}, ψ)$ of (8)–(9) by $\begin{matrix} (16) & \begin{matrix} E (t) = E (t) + \int_{0}^{L} F (ψ (t)) d x \end{matrix} \end{matrix}$ where $E (t)$ is the natural energy corresponding to the system (8)–(9) given by $\begin{matrix} (17) & \begin{matrix} E (t) = \frac{1}{2} {‖ (φ (t), φ_{t} (t), φ_{t t} (t), ψ (t)) ‖}_{H}^{2} . \end{matrix} \end{matrix}$

Lemma 2.1.
The total energy defined by ( 16 ) satisfies $\begin{matrix} (18) & \begin{matrix} \frac{d}{d t} E (t) = - μ ‖ ψ_{t} ‖_{2}^{2} . \end{matrix} \end{matrix}$ Moreover, there exist constants $C_{1}, C_{2} > 0$ such that $\begin{matrix} (19) & \begin{matrix} E (t) - C_{1} ⩽ E (t) ⩽ C_{2} (1 + {(E (t))}^{θ + 1}), \forall t ⩾ 0 . \end{matrix} \end{matrix}$
Proof.
A straightforward computation yields (18) by multiplying (formally) the equations (8) by $φ_{t}$ and (9) by $ψ_{t}$ , respectively. Finally, the estimate (19) follows easily by (14) and (15). The proof is complete. □

2.2. Well-posedness

In order to state our main result, we begin with a precise definition of a weak solution to (8)–(11).

Definition 2.1.
Given an initial data $(φ_{0}, φ_{1}, φ_{2}, ψ_{0}) \in H$ , a function $z = (φ, φ_{t}, φ_{t t}, ψ) \in C ([0, T]; H)$ is said to be a weak solution of (8)–(11) if for almost everywhere $t \in [0, T]$ , $\begin{array}{l} (20) & ρ_{1} \frac{d}{d t} (φ_{t}, u) + κ (φ_{x} + ψ, u_{x}) = 0, \\ (21) & ρ_{1} \frac{d}{d t} (φ_{t t}, v) + κ \frac{d}{d t} (φ_{x} + ψ, v_{x}) = 0, \\ (22) & ρ_{2} \frac{d}{d t} (φ_{t}, w_{x}) + b (ψ_{x}, w_{x}) + κ (φ_{x} + ψ, w) + μ (ψ_{t}, w) + (f (ψ), w) = 0, \end{array}$ for all $u, v \in H_{0}^{1} (0, L)$ , $w \in H_{}^{1} (0, L)$ and $(φ (0), φ_{t} (0), φ_{t t} (0), ψ (0)) = (φ_{0}, φ_{1}, φ_{2}, ψ_{0})$ .
Theorem 2.2.
Suppose that assumptions (* 14 )–( 15 ) hold. Then, we have

(i) If the initial data $(φ_{0}, φ_{1}, φ_{2}, ψ_{0}) \in H$ , then the system ( 8 )–( 11 ) has a weak solution satisfying $\begin{array}{l} φ \in L^{\infty} (0, T; H_{0}^{1} (0, L)), ψ \in L^{\infty} (0, T; H_{}^{1} (0, L)), \\ φ_{t} \in L^{\infty} (0, T; H_{0}^{1} (0, L)), φ_{t t} \in L^{\infty} (0, T, L^{2} (0, L)) . \end{array}$

(ii) If the initial data* $(φ_{0}, φ_{1}, φ_{2}, ψ_{0}) \in H_{1}$ , then the system ( 8 )–( 11 ) has a unique stronger weak solution satisfying $\begin{array}{l} φ \in L^{\infty} (0, T, H^{2} (0, L) \cap H_{0}^{1} (0, L)), ψ \in L^{\infty} (0, T, H_{}^{2} (0, L)), \\ φ_{t} \in L^{\infty} (0, T, H^{2} (0, L) \cap H_{0}^{1} (0, L)), φ_{t t} \in L^{\infty} (0, T, H_{0}^{1} (0, 1)) . \end{array}$

(iii) In both cases, the solution* $(φ, φ_{t}, φ_{t t}, ψ)$ depends continuously on the initial data in $H$ . In particular, the system ( 8 )–( 11 ) has a unique weak solution.
Proof.
The proof is given by the Faedo–Galerkin method.

Step 1 – Approximate problem. Let us consider initial data $(φ_{0}, φ_{1}, φ_{2}, ψ_{0}) \in H$ . Let ${ω_{j}}_{j = 1}^{\infty}$ and ${μ_{j}}_{j = 1}^{\infty}$ basis formed by the eigenfunctions of $- \partial_{x x}$ . It is well known that these bases can be considered orthogonal in $H^{2} (0, L) \cap H_{0}^{1} (0, L)$ and $H_{}^{2} (0, L)$ , respectively, and both orthonormal in $L^{2} (0, L)$ . Now we consider the finite-dimensional subspaces $\begin{matrix} H_{n} = span {ω_{1}, ω_{2}, \dots, ω_{n}}, V_{n} = span {μ_{1}, μ_{2}, \dots, μ_{n}}, n \in N . \end{matrix}$ We will find an approximate solution of the form $\begin{matrix} (23) & φ^{n} (x, t) = \sum_{j = 1}^{n} a_{j, n} ω_{j} (x), ψ^{n} (x, t) = \sum_{j = 1}^{n} b_{j, n} μ_{j} (x), \end{matrix}$ to the following approximate problem $\begin{array}{l} (24) & ρ_{1} (φ_{t t}^{n}, u) + κ (φ_{x}^{n} + ψ^{n}, u_{x}) = 0, \\ (25) & ρ_{1} (φ_{t t t}^{n}, v) + κ (φ_{x t}^{n} + ψ_{t}^{n}, v_{x}) = 0, \\ (26) & ρ_{2} (φ_{t t}^{n}, w_{x}) + b (ψ_{x}^{n}, w_{x}) + κ (φ_{x}^{n} + ψ^{n}, w) + μ (ψ_{t}^{n}, w) + (f (ψ^{n}), w) = 0, \end{array}$ for all $u, v \in H_{n}$ , $w \in V_{n}$ with initial conditions such that $\begin{matrix} (27) & (φ^{n} (0), φ_{t}^{n} (0), φ_{t t}^{n} (0) ψ^{n} (0)) = (φ_{0}^{n}, φ_{1}^{n}, φ_{2}^{n}, ψ_{0}^{n}) \to (φ_{0}, φ_{1}, φ_{2}, ψ_{0}) strongly in H . \end{matrix}$ From the application of the standard ODE theory, one has a local solution $(φ^{n} (t), φ_{t}^{n} (t), φ_{t t}^{n} (t), ψ^{n} (t))$ on the maximal interval $[0, t_{n})$ with $0 < t_{n} ⩽ T$ for every $n \in N$ .

Step 2 – A prior estimate. Replacing u by $φ_{t}^{n}$ in (24), v by $φ_{t t}^{n}$ in (25) and w by $ψ_{t}^{n}$ in (26), we obtain $\begin{array}{rcl} (28) & \frac{d}{d t} E^{n} (t) = - μ {‖ ψ_{t}^{n} ‖}_{2}^{2}, \end{array}$ where $\begin{matrix} (29) & \begin{matrix} E^{n} (t) : = E^{n} (t) + \int_{0}^{L} F (ψ^{n}) d x and E^{n} (t) : = \frac{1}{2} {‖ (φ^{n} (t), φ_{t}^{n} (t), φ_{t t}^{n} (t), ψ^{n} (t)) ‖}_{H}^{2} . \end{matrix} \end{matrix}$ Thus, using the second inequality in (19) and integrating (28) from 0 to $t < t_{n}$ , we obtain from our choice of initial data that for all $t \in [0, T]$ and for every $n \in N$ , $\begin{array}{rcl} (30) & E^{n} (t) + μ \int_{0}^{t} {‖ ψ_{t}^{n} (s) ‖}_{2}^{2} d s = E^{n} (0) ⩽ C_{2} (1 + {(E^{n} (0))}^{θ + 1}) ⩽ R_{1}, \end{array}$ where $R_{1}$ is a positive constant independent n. Thus, using the first inequality in (19) yields $\begin{matrix} E^{n} (t) + μ \int_{0}^{t} {‖ ψ_{t}^{n} (s) ‖}_{2}^{2} d s ⩽ C_{1} + R_{1}, \forall t ⩾ 0 . \end{matrix}$ Therefore, approximate solutions are defined on the whole range $[0, T]$ .

Step 3 – Passage to the limit. From (30) and definition of $E^{n} (t)$ , we deduce that $\begin{matrix} \{\begin{matrix} {φ^{n}} & is bounded in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ {φ_{t}^{n}} & is bounded in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ {φ_{t t}^{n}} & is bounded in L^{\infty} (0, T; L^{2} (0, L)), \\ {ψ^{n}} & is bounded in L^{\infty} (0, T; H_{}^{1} (0, L)), \\ {ψ_{t}^{n}} & is bounded in L^{2} (0, T; L^{2} (0, L)) . \end{matrix} \end{matrix}$ Therefore, we can extract subsequences of ${φ^{n}}$ and ${ψ^{n}}$ still denoted by ${φ^{n}}$ and ${ψ^{n}}$ such that $\begin{matrix} \{\begin{matrix} φ^{n} \to φ & weakly star in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ φ_{t}^{n} \to φ_{t} & weakly star in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ φ_{t t}^{n} \to φ_{t t} & weakly star in L^{\infty} (0, T; L^{2} (0, L)), \\ ψ^{n} \to ψ & weakly star in L^{\infty} (0, T; H_{}^{1} (0, L)), \\ ψ_{t}^{n} \to ψ_{t} & weakly star in L^{2} (0, T; L^{2} (0, L)) . \end{matrix} \end{matrix}$ By using Aubin–Lions lemma [22], we arrive at $\begin{array}{l} (31) & φ^{n} \to φ strongly in C ([0, T]; L^{2} (0, L)), \\ (32) & φ_{t}^{n} \to φ_{t} strongly in C ([0, T]; L^{2} (0, L)), \\ (33) & ψ^{n} \to ψ strongly in C ([0, T]; L^{2} (0, L)) . \end{array}$ Thus, using (14), the embedding $H^{1} (0, L) ↪ L^{\infty} (0, L)$ and the Hölder’s inequality, we obtain $\begin{matrix} (f (ψ^{n}) - f (ψ), w) & ⩽ k_{0} \int_{0}^{L} (1 + {| ψ^{n} |}^{θ - 1} + | ψ |^{θ - 1}) | ψ^{n} - ψ | | w | d x \\ ⩽ C ‖ w ‖_{2} {‖ ψ^{n} - ψ ‖}_{2} . \end{matrix}$ Then the convergence (33) gives $\begin{matrix} lim_{n \to \infty} (f (ψ^{n}), w) = (f (ψ), w) . \end{matrix}$ Therefore the above limits allows us to pass to the limit in the approximate problem (24)–(26) to get a weak solution satisfying $\begin{array}{rcl} φ \in L^{\infty} (0, T; H_{0}^{1} (0, L)), ψ \in L^{\infty} (0, T; H_{}^{1} (0, L)), \\ φ_{t} \in L^{\infty} (0, T; H_{0}^{1} (0, L)), φ_{t t} \in L^{\infty} (0, T, L^{2} (0, L)) . \end{array}$

Step 4 – Initial data. Firstly, we observe that (31)–(33) implies $\begin{matrix} (φ (0), φ_{t} (0), ψ (0)) = (φ_{0}, φ_{1}, ψ_{0}) . \end{matrix}$ Now, we multiply (25) by a test function $\begin{matrix} η \in H^{1} (0, T), η (0) = 1, η (T) = 0, \end{matrix}$ and integrate the result over $[0, T]$ to obtain $\begin{matrix} - ρ_{1} (φ_{2}^{n}, v) - ρ_{1} \int_{0}^{T} (φ_{t t}^{n}, v) η_{t} d t + κ \int_{0}^{T} \frac{d}{d t} (φ_{x}^{n} + ψ^{n}, v_{x}) η d t = 0, \end{matrix}$ for all $v \in H_{0}^{1} (0, L)$ . Taking the limit as $n \to \infty$ , we obtain $\begin{matrix} (34) & - ρ_{1} (φ_{2}, v) - ρ_{1} \int_{0}^{T} (φ_{t t}, v) η_{t} d t + κ \int_{0}^{T} \frac{d}{d t} (φ_{x} + ψ, v_{x}) η d t = 0, \end{matrix}$ for all $v \in H_{0}^{1} (0, L)$ . On the other hand, multiplying (21) by η and integrating the result over $[0, T]$ , we obtain $\begin{matrix} (35) & - ρ_{1} (φ_{t t} (0), v) - ρ_{1} \int_{0}^{T} (φ_{t t}, v) η_{t} d t + κ \int_{0}^{T} \frac{d}{d t} (φ_{x} + ψ, v_{x}) η d t = 0, \end{matrix}$ for all $v \in H_{0}^{1} (0, L)$ . Combining (34) and (35) we conclude that $φ_{t t} (0) = φ_{2}$ .

Step 5 – Stronger solutions. Suppose that the initial data in the approximate problem (24)–(26) satisfies $(φ_{0}, φ_{1}, φ_{2}, ψ_{0}) \in H_{1}$ and $\begin{matrix} (36) & (φ_{0}^{n}, φ_{1}^{n}, φ_{2}^{n}, ψ_{0}^{n}) \to (φ_{0}, φ_{1}, φ_{2}, ψ_{0}) strongly in H_{1} . \end{matrix}$ Replacing u by $- φ_{x x t}^{n}$ in (24), v by $- φ_{x x t t}^{n}$ in (25) and w by $- ψ_{x x t}^{n}$ in (26), we see that $\begin{matrix} (37) & \frac{d}{d t} F^{n} (t) + μ {‖ ψ_{x t}^{n} ‖}_{2}^{2} = - (f (ψ^{n}), ψ_{x x t}^{n}), \end{matrix}$ where $\begin{array}{rcl} F^{n} (t) : = \frac{ρ_{1}}{2} {‖ φ_{x t}^{n} ‖}_{2}^{2} + \frac{ρ_{1} ρ_{2}}{2 κ} {‖ φ_{x t t}^{n} ‖}_{2}^{2} + \frac{ρ_{2}}{2} {‖ φ_{x x t}^{n} ‖}_{2}^{2} + \frac{b}{2} {‖ ψ_{x x}^{n} ‖}_{2}^{2} + \frac{κ}{2} {‖ φ_{x x}^{n} + ψ_{x}^{n} ‖}_{2}^{2} . \end{array}$ Using the embedding $H^{1} (0, L) ↪ L^{\infty} (0, L)$ , Hölder’s and Young’s inequalities and (14), we get that $\begin{matrix} (f (ψ^{n}), ψ_{x x t}^{n}) & = - \int_{0}^{L} f^{'} (ψ^{n}) ψ_{x}^{n} ψ_{x t}^{n} d x \\ ⩽ C \int_{0}^{L} (1 + {| ψ^{n} |}^{θ - 1}) | ψ_{x}^{n} | | ψ_{x t}^{n} | d x \\ ⩽ C (1 + {‖ ψ_{x}^{n} ‖}_{2}^{θ - 1}) {‖ ψ_{x}^{n} ‖}_{2} {‖ φ_{x t}^{n} ‖}_{2} \\ ⩽ C_{0} ({‖ ψ_{x}^{n} ‖}_{2}^{2} + {‖ ψ_{x t}^{n} ‖}_{2}^{2}) ⩽ C_{0} F^{n} (t) . \end{matrix}$ Inserting the last estimate into (37) yields $\begin{matrix} \frac{d}{d t} F^{n} (t) + μ {‖ ψ_{x t}^{n} ‖}_{2}^{2} ⩽ C_{0} F^{n} (t) . \end{matrix}$ Then, Gronwall’s inequality implies $\begin{array}{rcl} (38) & F^{n} (t) + μ \int_{0}^{t} {‖ ψ_{x t}^{n} (s) ‖}_{2}^{2} d s ⩽ R_{2}, \forall t \in [0, T], \end{array}$ where $R_{2}$ is a positive constant independent of t and n but depending on initial data and then we can conclude that $\begin{matrix} \{\begin{matrix} {φ^{n}} & is bounded in L^{\infty} (0, T; H^{2} (0, L) \cap H_{0}^{1} (0, L)), \\ {φ_{t}^{n}} & is bounded in L^{\infty} (0, T; H^{2} (0, L) \cap H_{0}^{1} (0, L)), \\ {φ_{t t}^{n}} & is bounded in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ {ψ^{n}} & is bounded in L^{\infty} (0, T; H_{}^{2} (0, L)), \\ {ψ_{t}^{n}} & is bounded in L^{2} (0, T; H_{}^{1} (0, L)) . \end{matrix} \end{matrix}$ This implies that $\begin{matrix} \{\begin{matrix} φ^{n} \to φ & weakly star in L^{\infty} (0, T; H^{2} (0, L) \cap H_{0}^{1} (0, L)), \\ φ_{t}^{n} \to φ_{t} & weakly star in L^{\infty} (0, T; H^{2} (0, L) \cap H_{0}^{1} (0, L)), \\ φ_{t t}^{n} \to φ_{t t} & weakly star in L^{\infty} (0, T; H_{0}^{1} (0, L)), \\ ψ^{n} \to ψ & weakly star in L^{\infty} (0, T; H_{}^{2} (0, L)), \\ ψ_{t}^{n} \to ψ_{t} & weakly star in L^{2} (0, T; H_{}^{1} (0, L)) . \end{matrix} \end{matrix}$ From above limits we conclude that $(φ, φ_{t}, φ_{t t}, ψ)$ is a strong weak solution satisfying $\begin{array}{l} φ \in L^{\infty} (0, T, H^{2} (0, L) \cap H_{0}^{1} (0, L)), ψ \in L^{\infty} (0, T, H_{*}^{2} (0, L)), \\ φ_{t} \in L^{\infty} (0, T, H^{2} (0, L) \cap H_{0}^{1} (0, L)), φ_{t t} \in L^{\infty} (0, T, H_{0}^{1} (0, 1)) . \end{array}$

Step 6 – Continuous dependence. We consider the case of stronger solutions. Let $z^{1} (t) = (φ, φ_{t}, φ_{t t}, ψ)$ and $z^{2} (t) = (\tilde{φ}, {\tilde{φ}}_{t}, {\tilde{φ}}_{t t}, \tilde{ψ})$ be the stronger weak solutions of the problem (8)–(11) corresponding to initial data $z^{1} (0) = (φ_{0}, φ_{1}, φ_{2}, ψ_{0})$ , $z^{2} (0) = ({\tilde{φ}}_{0}, \tilde{φ_{1}}, \tilde{φ_{2}}, {\tilde{ψ}}_{0}) \in H_{1}$ , respectively. Then $(Φ, Φ_{t}, Φ_{t t}, Ψ) = z^{1} (t) - z^{2} (t)$ satisfies the following equations: $\begin{array}{l} (39) & ρ_{1} Φ_{t t} - κ {(Φ_{x} + Ψ)}_{x} = 0, \\ (40) & - ρ_{2} Φ_{t t x} - b Ψ_{x x} + κ (Φ_{x} + Ψ) + μ Ψ_{t} + f (ψ) - f (\tilde{ψ}) = 0, \end{array}$ with initial data $(Φ (0), Φ_{t} (0), Φ_{t t} (0), Ψ (0)) = z^{1} (0) - z^{2} (0)$ . Then, multiplying (39) by $Φ_{t}$ and (40) by $Ψ_{t}$ we obtain the energy dissipation law given by $\begin{array}{rcl} (41) & \frac{d}{d t} \hat{E} (t) = - μ ‖ Ψ_{t} ‖_{2}^{2} - (f (ψ) - f (\tilde{ψ}), Ψ_{t}), \end{array}$ where $\hat{E} (t)$ is the energy corresponding to $z^{1} (t) - z^{2} (t)$ defined by $\begin{matrix} (42) & \hat{E} (t) : = \frac{1}{2} {‖ (Φ, Φ_{t}, Φ_{t t}, Ψ) ‖}_{H}^{2} . \end{matrix}$ Using the embedding $H^{1} (0, L) ↪ L^{\infty} (0, L)$ , Hölder’s and Young’s inequalities and (14), we have $\begin{matrix} (43) & \begin{matrix} (f (ψ) - f (\tilde{ψ}), Ψ_{t}) & = \int_{0}^{L} (f (ψ) - f (\tilde{ψ})) Ψ_{t} d x \\ ⩽ C \int_{0}^{L} (1 + | ψ |^{θ - 1} + | \tilde{ψ} |^{θ - 1}) | Ψ | | Ψ_{t} | d x \\ ⩽ C (1 + ‖ ψ_{x} ‖_{2}^{θ - 1} + ‖ {\tilde{ψ}}_{x} ‖_{2}^{θ - 1}) ‖ Ψ ‖_{2} ‖ Ψ_{t} ‖_{2} \\ ⩽ C_{0} ‖ Ψ ‖_{2}^{2} + μ ‖ Ψ_{t} ‖_{2}^{2}, \\ ⩽ C_{0} \hat{E} (t) + μ ‖ Ψ_{t} ‖_{2}^{2}, \end{matrix} \end{matrix}$ where $\begin{matrix} C_{0} = C (1 + {‖ z^{1} ‖}_{H}^{2 (θ - 1)} + {‖ z^{2} ‖}_{H}^{2 (θ - 1)}), \end{matrix}$ and $C > 0$ is independent of $z^{1}$ and $z^{2}$ . Inserting the estimate (43) in (41), we conclude that $\begin{matrix} \frac{d}{d t} \hat{E} (t) ⩽ C_{0} \hat{E} (t), \end{matrix}$ and Gronwall’s lemma yields $\begin{matrix} (44) & \begin{matrix} {‖ z^{1} (t) - z^{2} (t) ‖}_{H}^{2} ⩽ C_{T} {‖ z^{1} (0) - z^{2} (0) ‖}_{H}^{2}, \forall t \in [0, T], \end{matrix} \end{matrix}$ where we obtain the continuous dependence of stronger weak solutions on the initial data and then we know that the stronger weak solutions of problem (8)–(11) is unique. The continuous dependence and uniqueness for weak solutions can be proved by using density arguments. Combining the above analysis, we complete the proof of Theorem 2.2. □

3. Long-time dynamics

3.1. Main result

To describe the results, we introduce the concept of global attractor and fractal dimension (see, e.g., [13,14]). A global attractor for a dynamical system $(X, S (t))$ is a compact set $A$ of X which is invariant, i.e., $S (t) A = A$ for all $t ⩾ 0$ and uniformly attracting, i.e., $\begin{matrix} lim_{t \to \infty} {dist}_{X} (S (t) B, A) = 0, \end{matrix}$ for any bounded set $B \subset X$ , where ${dist}_{X}$ is the Hausdorff semi-distance in X. The fractal dimension of a compact set $A \subset X$ is defined by $\begin{matrix} {dim}_{f} A : = lim_{ϵ \to 0} sup \frac{ln n (A, ϵ)}{ln (1 / ϵ)}, \end{matrix}$ where $n (A, ϵ)$ is the minimal number of closed balls of radius ϵ which cover A.

Thanks the well-posedness of weak solutions of the Timoshenko system, the solution operator $S (t) : H \to H$ , where $\begin{matrix} S (t) z = (φ (t), φ_{t} (t), φ_{t t} (t), ψ (t)), \forall t ⩾ 0, \end{matrix}$ is the weak solution to (8)–(11) with initial data $z \in H$ , defines a $C_{0}$ -semigroup on $H$ . We denote the associated dynamical system by $(H, S (t))$ .

The main result for the long-time dynamics is given by the following theorem.

Theorem 3.1.
Suppose that assumptions ( 14 )–( 15 ) hold. Then,
The dynamical system $(H, S (t))$ possesses a unique compact global attractor $A \subset H$ which is characterized by the unstable manifold $A = M_{+} (N)$ of the set of stationary solutions $\begin{matrix} N = \{(φ, 0, 0, ψ) \in H | \begin{matrix} - κ {(φ_{x} + ψ)}_{x} = 0 \\ - b ψ_{x x} + κ (φ_{x} + ψ) + f (ψ) = 0 \end{matrix}\} . \end{matrix}$

The global attractor $A$ has finite fractal dimension.

The global attractor $A$ is bounded in $H_{1} = {(H^{2} (0, L) \cap H_{0}^{1} (0, L))}^{2} \times H_{0}^{1} (0, L) \times H_{}^{2} (0, L)$ . Moreover, every trajectory* $z = (φ, φ_{t}, φ_{t t}, ψ)$ in $A$ satisfies $\begin{matrix} (45) & ‖ φ ‖_{H^{2} \cap H_{0}^{1}}^{2} + ‖ φ_{t} ‖_{H^{2} \cap H_{0}^{1}}^{2} + ‖ φ_{t t} ‖_{H_{0}^{1}}^{2} + ‖ φ_{t t t} ‖_{L^{2}} + ‖ ψ ‖_{H^{2} \cap H_{}^{1}}^{2} + ‖ ψ_{t} ‖_{H_{}^{1}}^{2} ⩽ R_{1}^{2}, \end{matrix}$ for some constant $R_{1} > 0$ dependent of $A$ .

Theorem 3.1 will be proved in the next subsections. The main ingredients of the proof are two: gradient systems and quasi-stability theory [13,14].
3.2. Gradient system

We recall that a dynamical system $(X, S (t))$ is gradient if it possesses a strict Lyapunov functional. That is, a functional $Φ : X \to R$ is a strict Lyapunov function for a system $(X, S (t))$ if,

the map $t \to Φ (S (t) z)$ is non-increasing for each $z \in X$ ,

if $Φ (S (t) z) = Φ (z)$ for some $z \in X$ and for all $t > 0$ , then z is a stationary point.

Lemma 3.2.
Suppose that assumptions ( 14 )–( 15 ) hold. Then the dynamical system $(H, S (t))$ is gradient, that is, there exists a strict Lyapunov function Φ defined in $H$ . In addition, $\begin{matrix} (46) & \begin{matrix} Φ (z) \to \infty if and only if ‖ z ‖_{H} \to \infty . \end{matrix} \end{matrix}$
Proof.
Let $z_{0} \in H$ and we call $S (t) z_{0} = (φ (t), φ_{t} (t), φ_{t t} (t), ψ (t))$ . By a density argument, we can assume that $S (t) z_{0}$ is a strong solution. Let us define the function $Φ : H \to R$ by $\begin{matrix} (47) & \begin{matrix} Φ (S (t) z_{0}) : = E (t) = \frac{1}{2} {‖ (φ (t), φ_{t} (t), φ_{t t} (t), ψ (t)) ‖}_{H}^{2} + \int_{0}^{L} F (ψ (t)) d x . \end{matrix} \end{matrix}$ Then by (18) we obtain that $\begin{matrix} (48) & \begin{matrix} \frac{d}{d t} Φ (S (t) z_{0}) = - μ ‖ ψ_{t} ‖_{2}^{2} ⩽ 0 . \end{matrix} \end{matrix}$ Thus, $t \mapsto Φ (S (t) z_{0})$ is a non-increasing function.

Suppose that $Φ (S (t) z_{0}) = Φ (z_{0})$ for some $z_{0} \in H$ and for all $t ⩾ 0$ . Then $\begin{matrix} - μ ‖ ψ_{t} ‖_{2}^{2} = \frac{d}{d t} Φ (z_{0}) = 0, \forall t ⩾ 0 . \end{matrix}$ Thus, $ψ (x, t) = ψ (x)$ for all $t ⩾ 0$ . Then by (9), we find $\begin{matrix} - ρ_{2} φ_{t t x} + κ φ_{x} = b ψ_{x x} - κ ψ - f (ψ) = : h_{1} (x) . \end{matrix}$ Integrating the last equality over $[0, x]$ and using the boundary condition (11), we obtain that $\begin{matrix} (49) & \begin{matrix} - ρ_{2} φ_{t t} + κ φ = \int_{0}^{x} h_{1} (s) d s = : h_{2} (x) . \end{matrix} \end{matrix}$ By (8) we have $\begin{matrix} (50) & \begin{matrix} ρ_{1} φ_{t t} - κ φ_{x x} = κ ψ_{x} = : h_{3} (x) . \end{matrix} \end{matrix}$ Combining (49) and (50), we obtain $\begin{matrix} (51) & \begin{matrix} φ_{x x} - \frac{ρ_{1}}{ρ_{2}} φ = \frac{ρ_{1}}{ρ_{2} κ} h_{2} (x) + \frac{1}{κ} h_{3} (x) = : h_{4} (x) . \end{matrix} \end{matrix}$ Taking the derivative of (51) with respect to the variable t and defining $u = φ_{t}$ and $ρ = \frac{ρ_{1}}{ρ_{2}}$ yields $\begin{matrix} \{\begin{matrix} u_{x x} - ρ u = 0, & in (0, L) \times R^{+}, \\ u (0, t) = u (L, t) = 0, & in R^{+} . \end{matrix} \end{matrix}$ By uniqueness of above BVP, we conclude that $φ_{t} = u = 0$ . Thus, $φ (x, t) = φ (x)$ for all $t ⩾ 0$ . Then $z_{0} = (φ, 0, 0, ψ)$ is a stationary point of $(H, S (t))$ and Φ is a strict Lyapunov function. Finally, property (46) follows promptly from estimate (19). The proof is complete. □
Lemma 3.3.
Under assumptions of Theorem 3.1 , the set $N$ of the stationary points of $(H, S (t))$ is bounded in $H$ .
Proof.
Let $z \in N$ be arbitrary. We know that $z = (φ, 0, 0, ψ)$ and satisfies the system $\begin{matrix} (52) & \begin{matrix} \{\begin{matrix} - κ {(φ_{x} + ψ)}_{x} = 0, & in (0, L) \times (0, T), \\ - b ψ_{x x} + κ (φ_{x} + ψ) + f (ψ) = 0, & in (0, L) \times (0, T) . \end{matrix} \end{matrix} \end{matrix}$ Multiplying the first equation in (52) by φ and second by ψ, respectively, taking the sum and integrating over $(0, L)$ , we get $\begin{matrix} b ‖ ψ_{x} ‖_{2}^{2} + κ ‖ φ_{x} + ψ ‖_{2}^{2} & = - \int_{0}^{L} f (ψ) ψ d x . \end{matrix}$ Using (15), we conclude that $\begin{matrix} (53) & \begin{matrix} ‖ z ‖_{H}^{2} ⩽ 2 L k_{1}, \forall z \in N . \end{matrix} \end{matrix}$ The proof is complete. □

3.3. Quasistability estimate

In this section, we derive quasistability estimate which is the main tool in proving existence, finite dimensionality and smoothness of attractors [13,14].

Theorem 3.4.
Let $S (t) z^{i} = (φ^{i}, φ_{t}^{i}, φ_{t t}^{i}, ψ^{i})$ , $i = 1, 2$ , be two solutions taken from an positively invariant bounded set B. Let $(φ, φ_{t}, φ_{t t}, ψ) = (φ^{1} - φ^{2}, φ_{t}^{1} - φ_{t}^{2}, φ_{t t}^{1} - φ_{t t}^{2}, ψ^{1} - ψ^{2})$ . Then the following inequality holds: $\begin{array}{rcl} (54) & {‖ S (t) z^{1} - S (t) z^{2} ‖}_{H}^{2} ⩽ ϑ e^{- η t} ‖ z_{1} - z_{2} ‖_{H}^{2} + C_{B} sup_{s \in [0, t]} {‖ ψ (s) ‖}_{2 θ}^{2}, \end{array}$ for some constants $ϑ, η, C_{B} > 0$ .

The proof of Theorem 3.4 consists of several steps. We start with some preliminary lemmas.

First we note that the function $(φ, φ_{t}, φ_{t t}, ψ)$ satisfies $\begin{array}{l} (55) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0 in (0, L) \times (0, \infty), \\ (56) & - ρ_{2} φ_{t t x} - b ψ_{x x} + κ (φ_{x} + ψ) + μ ψ_{t} + f (ψ^{1}) - f (ψ^{2}) = 0 in (0, L) \times (0, \infty), \end{array}$ with the initial conditions $\begin{array}{rcl} (57) & (φ (0), φ_{t} (0), φ_{t t} (0), ψ (0)) = z^{1} - z^{2}, \end{array}$ and boundary conditions of Dirichlet–Neumann–Dirichlet $\begin{matrix} (58) & φ (0, t) = φ (L, t) = ψ_{x} (0, t) = ψ_{x} (L, t) = 0, t ⩾ 0 . \end{matrix}$

First, we set the energy functional $\begin{array}{rcl} (59) & E (t) : = \frac{1}{2} {‖ (φ (t), φ_{t} (t), φ_{t t} (t), ψ (t)) ‖}_{H}^{2} . \end{array}$
Lemma 3.5.
The functional ( 59 ) satisfies the estimate $\begin{matrix} (60) & \begin{matrix} \frac{d}{d t} E (t) ⩽ - \frac{μ}{2} ‖ ψ_{t} ‖_{2}^{2} + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix} \end{matrix}$
Proof.
Multiplying (55) by $φ_{t}$ , (56) by $ψ_{t}$ , and integrating over $[0, L]$ , we obtain $\begin{matrix} (61) & \begin{matrix} \frac{d}{d t} E (t) = - μ ‖ ψ_{t} ‖_{2}^{2} - \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2}) ψ_{t} d x . \end{matrix} \end{matrix}$ By (14), Hölder’s inequality and Young’s inequality, we have $\begin{matrix} (62) & \begin{matrix} | \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2}) ψ_{t} d x | & ⩽ C (1 + {‖ ψ^{1} ‖}_{2 θ}^{θ - 1} + {‖ ψ^{2} ‖}_{2 θ}^{θ - 1}) ‖ ψ ‖_{2 θ} ‖ ψ_{t} ‖_{2} \\ ⩽ C_{B} ‖ ψ ‖_{2 θ} ‖ ψ_{t} ‖_{2} ⩽ C_{B} ‖ ψ ‖_{2 θ}^{2} + \frac{μ}{2} ‖ ψ_{t} ‖_{2}^{2} . \end{matrix} \end{matrix}$ Substituting the above estimate in (61) we conclude that (60) holds. The proof is complete. □

Next we introduce the functional $\begin{array}{rcl} (63) & F (t) : = - ρ_{1} \int_{0}^{L} φ_{t} φ d x . \end{array}$
Lemma 3.6.
For all $ε > 0$ , the functional ( 63 ) satisfies $\begin{array}{rcl} (64) & \frac{d}{d t} F (t) ⩽ - ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x + 2 ε c_{p} κ \int_{0}^{L} | ψ_{x} |^{2} d x + C_{ε} \int_{0}^{L} | φ_{x} + ψ |^{2} d x . \end{array}$
Proof.
Multiplying the equation (55) by φ, integrating over $(0, L)$ , using integration by parts and taking into account the boundary conditions (58), we have $\begin{array}{rcl} (65) & ρ_{1} \int_{0}^{L} φ_{t t} φ d x + κ \int_{0}^{L} (φ_{x} + ψ) φ_{x} d x = 0 . \end{array}$ Taking into account the identity $φ_{t t} φ = \frac{d}{d t} (φ_{t} φ) - | φ_{t} |^{2}$ and the Young’s inequality we arrive at $\begin{matrix} (66) & \begin{matrix} - \frac{d}{d t} (ρ_{1} \int_{0}^{L} φ_{t} φ d x) ⩽ & - ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x + \frac{κ}{4 ε} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + ε κ \int_{0}^{L} | φ_{x} |^{2} d x . \end{matrix} \end{matrix}$ Moreover, we consider the inequality given by $\begin{array}{rcl} (67) & \int_{0}^{L} | φ_{x} |^{2} d x ⩽ 2 \int_{0}^{L} | φ_{x} + ψ |^{2} d x + 2 c_{p} \int_{0}^{L} | ψ_{x} |^{2} d x, \end{array}$ where $c_{p} > 0$ is the Poincaré’s constant in order to rewritten the inequality (66) as $\begin{matrix} (68) & \begin{matrix} - \frac{d}{d t} (ρ_{1} \int_{0}^{L} φ_{t} φ d x) ⩽ & - ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x + 2 ε c_{p} κ \int_{0}^{L} | ψ_{x} |^{2} d x \\ + C_{ε} \int_{0}^{L} | φ_{x} + ψ |^{2} d x, \end{matrix} \end{matrix}$ where $C_{ε} : = 1 / 4 ε + 2 ε$ . □

We consider the functional $\begin{array}{rcl} (69) & G (t) : = ρ_{1} \int_{0}^{L} φ_{t} φ d x + \frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x . \end{array}$
Lemma 3.7.
The functional ( 69 ) satisfies $\begin{matrix} (70) & \begin{matrix} \frac{d}{d t} G (t) ⩽ & - \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x - κ \int_{0}^{L} | φ_{x} + ψ |^{2} d x + ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x \\ + \frac{ρ_{2}^{2}}{2 b} \int_{0}^{L} | φ_{t t} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix} \end{matrix}$
Proof.
Analogously to the previous lemma, we have $\begin{array}{rcl} (71) & \frac{d}{d t} (ρ_{1} \int_{0}^{L} φ_{t} φ d x) = ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x - κ \int_{0}^{L} (φ_{x} + ψ) φ_{x} d x . \end{array}$ Moreover, multiplying the equation (56) by ψ and integrating by parts we obtain $\begin{matrix} (72) & \begin{matrix} ρ_{2} \int_{0}^{L} φ_{t t} ψ_{x} d x + b \int_{0}^{L} | ψ_{x} |^{2} d x + κ \int_{0}^{L} (φ_{x} + ψ) ψ d x \\ + \frac{d}{d t} (\frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x) + \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2})) ψ d x = 0 . \end{matrix} \end{matrix}$ By (14), Hölder’s inequality and the embedding $L^{2 θ} (0, L) ↪ L^{2} (0, L)$ , it follows that $\begin{matrix} (73) & \begin{matrix} | \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2}) ψ d x | & ⩽ C (1 + {‖ ψ^{1} ‖}_{2 θ}^{θ - 1} + {‖ ψ^{2} ‖}_{2 θ}^{θ - 1}) ‖ ψ ‖_{2 θ} ‖ ψ ‖_{2} \\ ⩽ C_{B} ‖ ψ ‖_{2 θ} ‖ ψ ‖_{2} ⩽ C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix} \end{matrix}$ Inserting the estimate (73) into (72) and using the Young’s inequality we arrive at $\begin{matrix} (74) & \begin{matrix} \frac{d}{d t} (\frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x) ⩽ & \frac{ρ_{2}^{2}}{2 b} \int_{0}^{L} | φ_{t t} |^{2} d x - \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x \\ - κ \int_{0}^{L} (φ_{x} + ψ) ψ d x + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix} \end{matrix}$ Adding up (71) and (74) we conclude $\begin{matrix} \frac{d}{d t} (ρ_{1} \int_{0}^{L} φ_{t} φ d x + \frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x) ⩽ & \frac{ρ_{2}^{2}}{2 b} \int_{0}^{L} | φ_{t t} |^{2} d x + ρ_{1} \int_{0}^{L} | φ_{t} |^{2} d x - \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x \\ - κ \int_{0}^{L} | φ_{x} + ψ |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{matrix}$ from where we finished the proof of this lemma. □

Let us introduce the functional $\begin{array}{rcl} (75) & H (t) : = ρ_{2} \int_{0}^{L} φ_{x t} φ_{x} d x + \frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x . \end{array}$
Lemma 3.8.
The functional ( 75 ) satisfies $\begin{matrix} \frac{d}{d t} H (t) ⩽ & - \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} | φ_{t t} |^{2} d x - \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x + \frac{κ^{2} c_{p}}{2 b} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + ρ_{2} \int_{0}^{L} | φ_{t x} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{matrix}$ where $c_{p}$ is the Poincaré’s constant.
Proof.
It follows from equation (55) that $ψ_{x} = \frac{ρ_{1}}{κ} φ_{t t} - φ_{x x}$ . Then, substituting $ψ_{x}$ into (72) we obtain $\begin{matrix} \frac{d}{d t} (ρ_{2} \int_{0}^{L} φ_{x t} φ_{x} d x + \frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x) - ρ_{2} \int_{0}^{L} | φ_{t x} |^{2} d x + \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} | φ_{t t} |^{2} d x \\ + b \int_{0}^{L} | ψ_{x} |^{2} d x + κ \int_{0}^{L} (φ_{x} + ψ) ψ d x + \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2})) ψ d x = 0, \end{matrix}$ and using (73), Young’s and Poincare’s inequalities we arrive at $\begin{matrix} \frac{d}{d t} (ρ_{2} \int_{0}^{L} φ_{x t} φ_{x} d x + \frac{μ}{2} \int_{0}^{L} | ψ |^{2} d x) ⩽ & - \frac{ρ_{1} ρ_{2}}{κ} \int_{0}^{L} | φ_{t t} |^{2} d x - \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x \\ + ρ_{2} \int_{0}^{L} | φ_{t x} |^{2} d x + \frac{κ^{2} c_{p}}{2 b} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{matrix}$ where $c_{p} > 0$ is the Poincare’s constant. □

Let us introduce more one functional which is given by $\begin{array}{rcl} (76) & K (t) : = - ρ_{2} \int_{0}^{L} φ_{t x} (φ_{x} + ψ) d x - \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ d x . \end{array}$
Lemma 3.9.
The functional ( 76 ) satisfies $\begin{array}{rcl} (77) & \frac{d}{d t} K (t) ⩽ - \frac{ρ_{2}}{2} \int_{0}^{L} | φ_{t x} |^{2} d x - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x + C_{1} \int_{0}^{L} | ψ_{t} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{array}$ for some constant $C_{1} > 0$ .
Proof.
Multiplying the equation (56) by $(φ_{x} + ψ)$ , integrating over $(0, L)$ and using integration by parts we have $\begin{matrix} (78) & \begin{matrix} - ρ_{2} \int_{0}^{L} φ_{t t x} (φ_{x} + ψ) d x + b \int_{0}^{L} ψ_{x} {(φ_{x} + ψ)}_{x} d x + κ \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + μ \int_{0}^{L} ψ_{t} (φ_{x} + ψ) d x + \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2})) (φ_{x} + ψ) = 0 . \end{matrix} \end{matrix}$ Using (14) and Hölder’s inequality, yields $\begin{matrix} | \int_{0}^{L} (f (ψ^{1}) - f (ψ^{2}) (φ_{x} + ψ) d x | & ⩽ C (1 + {‖ ψ^{1} ‖}_{2 θ}^{θ - 1} + {‖ ψ^{2} ‖}_{2 θ}^{θ - 1}) ‖ ψ ‖_{2 θ} {‖ (φ_{x} + ψ) ‖}_{2} \\ ⩽ C_{B} ‖ ψ ‖_{2 θ} ‖ φ_{x} + ψ ‖_{2} \\ ⩽ C_{B} ‖ ψ ‖_{2 θ}^{2} + \frac{κ}{4} ‖ φ_{x} + ψ ‖_{2}^{2} . \end{matrix}$ Inserting the above estimate into (78) and using the Young’s inequality, it follows that $\begin{matrix} - ρ_{2} \int_{0}^{L} φ_{t t x} (φ_{x} + ψ) d x + b \int_{0}^{L} ψ_{x} {(φ_{x} + ψ)}_{x} d x ⩽ & - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x + \frac{μ^{2}}{κ} \int_{0}^{L} | ψ_{t} |^{2} d x \\ + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix}$ On the other hand, if follows from equation (55) that ${(φ_{x} + ψ)}_{x} = \frac{ρ_{1}}{κ} φ_{t t}$ and then we can rewritten the above inequality as $\begin{matrix} - ρ_{2} \int_{0}^{L} φ_{t t x} (φ_{x} + ψ) d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{t t} ψ_{x} d x ⩽ & - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x + \frac{μ^{2}}{κ} \int_{0}^{L} | ψ_{t} |^{2} d x \\ + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix}$ Moreover, we consider the identity given by $φ_{t t x} (φ_{x} + ψ) = \frac{d}{d t} [φ_{t x} (φ_{x} + ψ)] - φ_{t x} {(φ_{x} + ψ)}_{t}$ from where we obtain $\begin{matrix} - \frac{d}{d t} (ρ_{2} \int_{0}^{L} φ_{t x} (φ_{x} + ψ) d x) ⩽ & - ρ_{2} \int_{0}^{L} | φ_{t x} |^{2} d x - ρ_{2} \int_{0}^{L} φ_{t x} ψ_{t} d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{t t x} ψ d x \\ - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x + \frac{μ^{2}}{κ} \int_{0}^{L} | ψ_{t} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2} . \end{matrix}$ On the other hand, using the identity $φ_{t t x} ψ = \frac{d}{d t} (φ_{t x} ψ) - φ_{t x} ψ_{t}$ we have $\begin{matrix} - \frac{d}{d t} (ρ_{2} \int_{0}^{L} φ_{t x} (φ_{x} + ψ) d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ d x) \\ ⩽ - ρ_{2} \int_{0}^{L} | φ_{t x} |^{2} d x - b (\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b}) \int_{0}^{L} φ_{t x} ψ_{t} d x \\ - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x + \frac{μ^{2}}{κ} \int_{0}^{L} | ψ_{t} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{matrix}$ and using again the Young’s inequality we arrive at $\begin{matrix} - \frac{d}{d t} (ρ_{2} \int_{0}^{L} φ_{t x} (φ_{x} + ψ) d x + \frac{b ρ_{1}}{κ} \int_{0}^{L} φ_{x t} ψ d x) ⩽ & - \frac{ρ_{2}}{2} \int_{0}^{L} | φ_{t x} |^{2} d x - \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + C_{1} \int_{0}^{L} | ψ_{t} |^{2} d x + C_{B} ‖ ψ ‖_{2 θ}^{2}, \end{matrix}$ where $C_{1} = b^{2} {(\frac{ρ_{1}}{κ} + \frac{ρ_{2}}{b})}^{2} / 2 ρ_{2} + μ^{2} / κ$ . Therefore, we finished the proof of this lemma. □

Proof of Theorem 3.4 (completion). Now we are in position to proof the Theorem 3.4. In order to do this, we define the Lyapunov functional $L$ as follows $\begin{array}{rcl} (79) & L (t) : = N_{1} E (t) + N_{2} F (t) + G (t) + N_{3} H (t) + N_{4} K (t), \end{array}$ where $N_{i}$ , $i = 1, 2, 3, 4$ are positive constants to be fixed later and the functionals $F$ , $G$ , $H$ and $K$ are given in Lemmas 3.6, 3.7, 3.8 and 3.9, respectively. Then, taking into account the results of these lemmas and substituting them in the derivative of $L (t)$ we obtain that $\begin{matrix} \frac{d}{d t} L (t) ⩽ & - 2 (N_{2} - 1) \frac{ρ_{1}}{2} \int_{0}^{L} | φ_{t} |^{2} d x - (2 N_{3} - \frac{κ ρ_{2}}{b ρ_{1}}) \frac{ρ_{1} ρ_{2}}{2 κ} \int_{0}^{L} | φ_{t t} |^{2} d x \\ - (N_{4} - 2 N_{3}) \frac{ρ_{2}}{2} \int_{0}^{L} | φ_{x t} |^{2} d x - (1 + N_{3} - \frac{4 ε c_{p} κ}{b} N_{2}) \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x \\ - (2 + N_{4} - \frac{2 C_{ε}}{κ} N_{2} - \frac{κ c_{p}}{b} N_{3}) \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ - (\frac{μ}{2} N_{1} - C_{1} N_{4}) \int_{0}^{L} | ψ_{t} |^{2} d x + C_{B} (N_{1} + 1 + N_{4} + N_{5}) ‖ ψ ‖_{2 θ}^{2} . \end{matrix}$ Now, we will choose carefully the constants. First, we select $\begin{matrix} ε : = \frac{b}{8 κ c_{p} N_{2}} \end{matrix}$ and $N_{1}$ large enough such that $N_{1} > \frac{2 C_{1}}{μ} N_{4}$ from where we obtain $\begin{matrix} \frac{d}{d t} L (t) ⩽ & - 2 (N_{2} - 1) \frac{ρ_{1}}{2} \int_{0}^{L} | φ_{t} |^{2} d x - (2 N_{3} - \frac{κ ρ_{2}}{b ρ_{1}}) \frac{ρ_{1} ρ_{2}}{2 κ} \int_{0}^{L} | φ_{t t} |^{2} d x \\ - (N_{4} - 2 N_{3}) \frac{ρ_{2}}{2} \int_{0}^{L} | φ_{x t} |^{2} d x - (N_{3} + \frac{1}{2}) \frac{b}{2} \int_{0}^{L} | ψ_{x} |^{2} d x \\ - (2 + N_{4} - \frac{2 C_{ε}}{κ} N_{2} - \frac{κ c_{p}}{b} N_{3}) \frac{κ}{2} \int_{0}^{L} | φ_{x} + ψ |^{2} d x \\ + C_{B} (N_{1} + 1 + N_{4} + N_{5}) ‖ ψ ‖_{2 θ}^{2} . \end{matrix}$ Now, we choose $N_{2} > 1$ , $N_{3} > \frac{κ ρ_{2}}{2 b ρ_{1}}$ following by $N_{4}$ large enough such that $\begin{matrix} N_{4} > max {2 N_{3}, 2 \frac{C_{ε}}{κ} N_{2} + \frac{κ c_{p}}{b} N_{3}}, \end{matrix}$ from where we obtain that $\begin{array}{rcl} (80) & N_{4} - 2 N_{3} > 0 and 2 + N_{4} - \frac{2 C_{ε}}{κ} N_{2} - \frac{κ c_{p}}{b} N_{3} > 0 . \end{array}$ Now, we can conclude that there exists a positive constant $N_{0} > 0$ such that $\begin{array}{rcl} (81) & \frac{d}{d t} L (t) ⩽ - N_{0} E (t) + C_{B} ‖ ψ ‖_{2 θ}^{2}, \forall t ⩾ 0, \end{array}$ where $E (t)$ is given in (59). Now, using the fact that $L (t)$ is equivalent to $E (t)$ we get $\begin{array}{rcl} (82) & \frac{d}{d t} L (t) ⩽ - η L (0) + C_{B} ‖ ψ ‖_{2 θ}^{2}, \forall t ⩾ 0, \end{array}$ for some positive constant η. Thus, integrating the above inequality over $(0, t)$ , we arrive at $\begin{array}{rcl} (83) & L (t) ⩽ L (0) e^{- η t} + C_{B} \int_{0}^{t} e^{- η (t - s)} {‖ ψ (s) ‖}_{2 θ}^{2} d s . \end{array}$ Using again the equivalence between $L (t)$ and $E (t)$ , we conclude that there exists $ϑ > 0$ such that $\begin{matrix} E (t) ⩽ ϑ E (0) e^{- η t} + C_{B} sup_{s \in [0, t]} {‖ ψ (s) ‖}_{2 θ}^{2} . \end{matrix}$ Therefore, $\begin{matrix} {‖ S (t) z^{1} - S (t) z^{2} ‖}_{H}^{2} ⩽ ϑ e^{- η t} ‖ z_{1} - z_{2} ‖_{H}^{2} + C_{B} sup_{s \in [0, t]} {‖ ψ (s) ‖}_{2 θ}^{2} . \end{matrix}$ The proof is complete.
3.4. Proof of the main result

Proof of Theorem 3.1 . (a) We first prove that the dynamical system $(H, S (t))$ is asymptotically smooth using the “compensated compactness” result in [13, Theorem 2.2.17]. Let B be a bounded positively invariant set of $(H, S (t))$ . Thus, given $ϵ > 0$ and using the estimate (54) with $T > 0$ sufficiently large we obtain that $\begin{matrix} {‖ S (T) z^{1} - S (T) z^{2} ‖}_{H}^{2} ⩽ ϵ + Ψ_{B, T, ϵ} (z^{1}, z^{2}), \end{matrix}$ where $\begin{matrix} Ψ_{B, T, ϵ} (z^{1}, z^{2}) & = C_{B} sup_{τ \in [0, T]} {‖ ψ^{1} (τ) - ψ^{2} (τ) ‖}_{2 θ}^{2} . \end{matrix}$ Given a sequence of initial data $z^{n} \in B$ , we have that $S (t) z^{n} = (φ^{n} (t), φ_{t}^{n} (t), φ_{t t}^{n} (t), ψ^{n} (t))$ is uniformly bounded in $H$ by positive invariance of B. Hence, using the relations (18) and (19) we obtain $\begin{matrix} \int_{0}^{T} {‖ ψ_{t}^{n} (s) ‖}_{2}^{2} d s = - \frac{1}{μ} \int_{0}^{T} \frac{d}{d s} E^{n} (s) d s ⩽ \frac{2 | E^{n} (0) |}{μ} ⩽ C_{B}, \end{matrix}$ where $E^{n}$ is defined in (29). Consequently, $\begin{matrix} (84) & \begin{matrix} {ψ^{n}} are bounded in L^{\infty} (0, T; H_{*}^{1} (0, L)), \\ {ψ_{t}^{n}} are bounded in L^{2} (0, T; L^{2} (0, L)) . \end{matrix} \end{matrix}$ Using the Aubin–Lions theorem (see [30] Corollary 4) we conclude that $\begin{matrix} ψ^{n} \to ψ in C ([0, T]; L^{2 θ} (0, L)) . \end{matrix}$ This implies that $\begin{matrix} (85) & \begin{matrix} lim_{m \to \infty} lim_{n \to \infty} Ψ_{B, T, ϵ} (z^{n}, z^{m}) = 0 . \end{matrix} \end{matrix}$ Therefore, by [13, Theorem 2.2.17] the dynamical system is asymptotically smooth. Thus, since $(H, S (t))$ is also gradient by Lemma 3.2 with the corresponding Lyapunov functional satisfying (46) and the set of its stationary points $N$ is bounded in $H$ , the existence of a compact global attractor $A = M_{+}^{u} (N)$ is established by [14, Corollary 7.5.7].

(b) Because $H^{1} (0, L)$ is compactly embedded in $L^{2 θ} (0, L)$ , we can find $H_{*}$ such that $H$ is compactly embedded in $H_{*}$ and $\begin{matrix} (86) & \begin{matrix} ‖ ψ ‖_{2 θ}^{2} ⩽ {‖ (φ, φ_{t}, φ_{t t}, ψ) ‖}_{H_{*}}^{2} . \end{matrix} \end{matrix}$ Therefore, we can apply general abstract results in [14, Theorems 8.6.1 and 8.7.1] to conclude that the attractor $A$ is of finite fractal dimension and also smooth. This means that the trajectories on the attractor satisfy $\begin{matrix} φ_{t} \in L^{\infty} (R, H_{0}^{1} (0, L)) \cap C (R, L^{2} (0, L)), φ_{t t} \in L^{\infty} (R, H_{0}^{1} (0, L)), \\ φ_{t t t} \in L^{\infty} (R, L^{2} (0, L)), ψ_{t} \in L^{\infty} (R, H_{*}^{1} (0, L)) . \end{matrix}$ Using (8)–(9) and noting that the nonlinear term f is locally continuous we have $\begin{matrix} κ φ_{x x} = ρ_{1} φ_{t t} - κ ψ_{x} \in L^{\infty} (R, L^{2} (0, L)), \\ b ψ_{x x} = - ρ_{2} φ_{t t x} + κ (φ_{x} + ψ) + μ ψ_{t} + f (ψ) \in L^{\infty} (R, L^{2} (0, L)) . \end{matrix}$ Therefore, we conclude that $A$ is bounded in $H_{1}$ . This completes the proof of Theorem 3.1.

4. Statements and declarations

Funding. M. M. Freitas was partially supported by CNPq Grant 313081/2021-2. D. S. Almeida Júnior was partially supported by CNPq Grant 314273/2020-4. A. J. A. Ramos was partially supported by CNPq Grant 310729/2019-0.

Competing interests. The authors have no relevant financial or non-financial interests to disclose

Data availability. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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