Smooth attractor for a nonlinear thermoelastic diffusion thin plate based on Gurtin

Abstract

A nonlinear system of sixth-order evolution equations which takes into account the hereditary effects via Gurtin–Pipkin’s model and the rotational inertia is considered. The system describes the behavior of thermoelastic diffusion thin plates, recently derived by Aouadi [Applied Mathematics and Mechanics (English Edition) 36 (2015), 619–632], where the heat and diffusion fluxes depend on the past history of the temperature and diffusion gradients through memory kernels, respectively. We prove the existence and uniqueness of global solutions as well as the exponential stability of the linear system at a rate proportional to the rotational inertia parameter. The existence of a global attractor whose fractal dimension is finite is proved. Finally, a smoothness property of the attractor is established with respect to the rotational inertia parameter.

Keywords

thermoelastic diffusion plate Gurtin–Pipkin’s model exponential stability global attractor

1. Introduction

Our main goal in this paper is to study the asymptotic behavior of the following thermoelastic diffusion thin plate equations based on Gurtin–Pipkin’s model, derived recently by Aouadi [2] in the framework of Lagnese and Lions’ model $\begin{matrix} u_{t t} - ϖ Δ u_{t t} + m Δ^{2} u + γ_{1} Δ θ + γ_{2} Δ P + f (u) = 0 in Ω \times R^{+}, \\ c θ_{t} + d P_{t} - \int_{0}^{\infty} k (s) (Δ θ (t - s) - h θ (t - s)) d s - γ_{1} Δ u_{t} = 0 in Ω \times R^{+}, \\ d θ_{t} + r P_{t} - \int_{0}^{\infty} ℏ (s) (Δ P (t - s) - h P (t - s)) d s - γ_{2} Δ u_{t} = 0 in Ω \times R^{+}, \end{matrix}$ (1.1) with the boundary conditions $u = Δ u = 0 on Γ \times R^{+}, θ = P = 0 on Γ \times R,$ (1.2) and the initial conditions $\begin{matrix} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), \\ θ (x, t) |_{t ⩽ 0} = θ_{0} (x, - t), P (x, t) |_{t ⩽ 0} = P_{0} (x, - t), x \in Ω, \end{matrix}$ (1.3) where $θ_{0} (x, t)$ and $P_{0} (x, t)$ are prescribed for $t ⩽ 0$ .

Throughout the paper Ω is a bounded domain in $R^{2}$ with smooth boundary Γ, u represents the vertical displacement of the plate, while θ is the temperature variation from the equilibrium reference value, P is the chemical potential and h is the thickness of the plate. The parameter ϖ is proportional to the square of the thickness of the plate and is neglected in some models (i.e. $ϖ = 0$ ). In this paper we consider the case $ϖ > 0$ , which corresponds to taking into account the rotational inertia of filaments of the plate. All the other physical constants are positive and we assume that $c r - d^{2} > 0 .$ (1.4) Note that this condition implies that $c θ^{2} + 2 d θ P + r P^{2} > 0 .$ Condition (1.4) is needed to stabilize thermoelastic diffusion systems (see [3] for more information on this). By virtue of (1.4) we deduce that $d / c < r / d$ . Let, then, κ be a number chosen in such a way that $d / c < κ < r / d$ . Thus, the Young’s inequality leads to $2 d \int_{Ω} | θ P | d x ⩽ \frac{d}{κ} \int_{Ω} | θ |^{2} d x + d κ \int_{Ω} | P |^{2} d x .$ (1.5)

We may think that the classical theory of thermoelasticity is a good model to explain the thermal conduction in thin plates. However, the research conducted in the development of high technologies after the Second World War confirmed that the mass diffusion field in solids cannot be ignored. So, the obvious question is what happens when the diffusion effects are considered together with the thermal effects in the theory of thin elastic plates. Diffusion can be defined as the random walk of a set of particles from regions of high concentration to regions of lower concentration. Thermodiffusion in an elastic solid is due to the coupling of the strain, temperature and mass diffusion fields. The heat and mass diffusion processes play an important role in many engineering applications, such as satellite problems, returning space vehicles and aircraft landing on water or land. There is now a great deal of interest in the diffusion process in the manufacturing of integrated circuits, integrated resistors, semiconductor substrates and MOS transistors. Oil companies are also interested in this phenomenon to improve the conditions of oil extractions.

Taking into account the mass diffusion effects in the thermoelasticity theory makes (1.1) correspond to the coupling of three hyperbolic equations (instead of two in thermoelasticity). This poses some new mathematical difficulties due to the relevant coupling between temperature and chemical potential in particular. In addition, we study our problem in the case where heat and diffusion conductions are of Gurtin–Pipkin’s type. This model, unlike that based on Fourier’s law, describes the physical phenomena more accurately due to the presence of the convolution integrals which entail finite propagation speed of thermal and mass diffusion disturbances. It is easy to see that the convolution terms in the second and third equations of (1.1) can be reduced to the classical Fourier’s law if k and ℏ are taken as the Dirac mass at zero.

There is an extensive bibliography on thin domain problems especially devoted to thermoelastic plates with memory (see e.g. [5,9–19] and references therein). To the authors best knowledge, the long-time behavior of thermoelastic diffusion plates with the rotational inertia was not considered at all. The focus of this paper is the study of the long-time properties of the dynamical system generated by problem (1.1)–(1.3) in the natural weak energy phase space, a particular emphasis being placed on the dependence of the long-time characteristics with respect to the varying parameter $0 ⩽ ϖ ⩽ 1$ . This includes questions such as: (i) existence of a compact global attractor and its structure, (ii) finite dimensionality of the attractor and smoothness properties (with respect to the parameter ϖ). We shall also provide conditions on the physical parameters under which a smooth attractor exists. The main difficulty, in treating such a problem, is that, due to the presence of the memory terms (the time convolution of a linear operator applied to the unknown function with a suitable memory kernel), the system is nonlocal; furthermore, the values of the unknown functions are known for all negative times.

Let us make some preliminary historical and bibliographical remarks on the long-time dynamics for thin thermoelastic plates with memory. Firstly we consider the problem with rotational inertia effects ( $ϖ > 0$ ). Bounds on the size of the attractor were first obtained in [6] for a linearly damped von Karman plates equation. Later, uniform bounds on the size and also the dimension of the attractor were established in [9] for a thermoelastic von Karman model, in the absence of mechanical dissipation. Grasselli and Squassina [19] proved, via energy methods, that the presence of rotational inertia restores the exponential stability of the linear thermoelastic plates with memory. Dell’Oro and Pata [12] studied the asymptotic properties of an abstract thermoelastic extensible beam or Berger plate of type III with respect to Gurtin–Pipkin’s model. Barbosa and Ma [5] proved that the thermal dissipation is sufficient to stabilize the system and guarantee the existence of a finite-dimensional global attractor and exponential attractors also. Using the $C_{0}$ -semigroup theory of linear operators, Aouadi proved in [2] the well-posedness of the thermoelastic diffusion plates equations with respect to the Fourier’s law. On the other hand, when the rotational inertia parameter is neglected ( $ϖ = 0$ ), Grasselli and Pata [18] proved that thermoelastic plates systems generate a strongly continuous semigroup $S (t)$ acting on an appropriate (extended) phase space such that any trajectory goes to zero as time goes to infinity. However, $S (t)$ fails to be exponentially stable [17]. This was also proved, for more general coupling terms, by Coti Zelati et al. [11]. Wu [29], by adding a structural damping of the form $Δ u_{t}$ , proved the existence of a global attractor and also the convergence of global solutions to equilibria. In the same direction, Potomkin [26] considered the nonlinear system and proved the existence of global and exponential attractors by adding a memory term. Recently Aouadi and Mairanville [4] proved the existence of a finite-dimensional global attractor for a nonlinear thermoelastic diffusion plates model with memory.

Motivated by the above results we propose to study the properties of global attractors to the problem (1.1)–(1.3) with only one nonlinear term and with the sole dissipation due to the thermal and chemical potential memories. This work is a continuation of [2] and [4] and complements the studies elaborated in the framework of thermoelastic thin plates with memory.

An absorbing set, besides giving a first rough estimate of the dissipativity of the system, is the preliminary step to prove the existence of much more interesting objects describing the asymptotic dynamics, such as global or exponential attractors (see, for instance, [21] and [23]). Unfortunately, if the dissipation is very weak, a direct proof of the existence of an absorbing set via explicit energy estimates might be very hard to find. On the other hand, for a quite general class of autonomous problems (the so-called gradient systems), it is possible to use an alternative approach and overcome this obstacle, appealing to the existence of a Lyapunov functional (see [21] and [22]). In that case, if the semigroup possesses suitable smoothing properties, one obtains right away the global attractor and the existence of an absorbing set is then recovered as a byproduct. To overcome the lack of compactness, our strategy is based on the existence of a Lyapunov functional, which reflects the gradient system structure of the problem, along with the exploitation of certain dissipation integrals and sharp energy estimates. The existence of the compact global attractor, its finite dimensionality and boundedness with respect to the rotational inertia parameter are obtained. The main technique for treating the model is the so-called stabilizability inequality derived from the exponential stability of the linear system of (1.1)–(1.3) (in our paper this inequality is formulated in Theorem 4.3). Similar inequalities were also obtained in various problems concerned with the dissipative wave dynamics and have become an important part of the study of the existence, smoothness and finite dimensionality of attractors (see [7–10] and references therein). We emphasize that these estimates in our case are not the consequences of some common abstract results, but are essentially determined by the physical parameters of our problem and depend on its peculiarities.

The well-known lack of regularity in the heat conduction equation of Gurtin–Pipkin’s type makes it quite difficult to handle already at the level of the well-posedness. Compared with the Coleman–Gurtin’s law (the heat equation contains the term $- ς Δ θ$ , where $ς > 0$ ), the dissipation in our system is only due to the memory effects of temperature and chemical potential, which is rather weak. The stronger dissipation provided in the Coleman–Gurtin’s law would make the problem easier to be dealt with.

This main novelties of the present paper are the following.

Some technical difficulties are found due to the combination of the nonlinear term f, the rotational inertia $ϖ Δ u_{t t}$ , θ and P. For this reason, instead of showing directly that the system has a bounded absorbing set, we show that it is gradient and asymptotically compact.

Only one damping term is considered in (1.1) to avoid some unrealistic dissipations.

The Gurtin–Pipkin’s law is considered instead of the Coleman–Gurtin’s law which gives stronger dissipation and is easier to be dealt with.

The decay rate of the linear system is shown to be proportional to the rotational inertia parameter. This will be achieved by using a multiplier technique based essentially on the construction of a suitable Lyapunov functional. Thus the presence of rotational inertia restores the exponential stability, a pleasant feature from the physical viewpoint.

The remaining part of this paper is organized as follows. In Section 2, we introduce the functional setting and we study the well-posedness of our problem. In Section 3 we prove that if $ϖ > 0$ , then the linear system generates an exponentially stable semigroup with a decay rate which is proportional to ϖ. In the final Section 4, we prove the existence of a global attractor with finite fractal dimension. Finally, the smoothness of the attractor is established with respect to ϖ.

2. Well-posedness

In this section we prove the well-posedness of the initial-boundary-value problem (1.1)–(1.3) by using the $C_{0}$ -semigroup theory of linear operators.

As mentioned in the Introduction, it is more convenient to work in the history space setting by introducing the so-called summed past history of θ and P defined by (cf. [1,15] and [16]) $η^{t} (s) = \int_{0}^{s} θ (t - τ) d τ, ν^{t} (s) = \int_{0}^{s} P (t - τ) d τ, (t, s) \in [0, \infty) \times R^{+} .$ (2.1) Differentiating (2.1)₁ and (2.1)₂ leads to further equations ruling the summed past history of θ and P with $η^{t}$ and $ν^{t}$ , respectively, $η_{t}^{t} (s) + η_{s}^{t} (s) = θ (t), ν_{t}^{t} (s) + ν_{s}^{t} (s) = P (t), (t, s) \in [0, \infty) \times R^{+},$ (2.2) subjected to the boundary and initial conditions $\begin{matrix} η^{t} (0) = ν^{t} (0) = 0 in Ω, t ⩾ 0, \\ η^{0} (s) = η_{0} (s) = \int_{0}^{s} θ_{0} (τ) d τ in Ω, s ⩾ 0, \\ ν^{0} (s) = ν_{0} (s) = \int_{0}^{s} P_{0} (τ) d τ in Ω, s ⩾ 0 . \end{matrix}$ (2.3)

Concerning the memory kernels k and ℏ, we set $μ (s) = - k^{'} (s), λ (s) = - ℏ^{'} (s),$ and we assume the following set of hypotheses on μ and λ

$μ, λ \in C^{1} (R^{+}) \cap L^{1} (R^{+})$ ,

$μ (s), λ (s) ⩾ 0$ $\forall s \in R^{+}$ ,

$μ^{'} (s), λ^{'} (s) ⩽ 0$ $\forall s \in R^{+}$ ,

$μ^{'} (s) + δ_{1} μ (s) ⩽ 0$ , $λ^{'} (s) + δ_{2} λ (s) ⩽ 0$ for some $δ_{1}, δ_{2} > 0$ , $\forall s \in R^{+}$ ,

$μ (0) = \int_{0}^{\infty} μ (s) d s : = μ_{0} > 0$ , $λ (0) = \int_{0}^{\infty} λ (s) d s : = λ_{0} > 0$ .

Then from (2.1) we have $\int_{0}^{\infty} k (s) θ (t - s) d s = - \int_{0}^{\infty} k^{'} (s) η^{t} (s) d s, \int_{0}^{\infty} ℏ (s) P (t - s) d s = - \int_{0}^{\infty} ℏ^{'} (s) ν^{t} (s) d s,$ and therefore $\int_{0}^{\infty} k (s) B θ (t - s) d s = \int_{0}^{\infty} μ (s) B η^{t} (s) d s, \int_{0}^{\infty} ℏ (s) B P (t - s) d s = \int_{0}^{\infty} λ (s) B ν^{t} (s) d s,$ where $B = h I - Δ$ . Consequently, problem (1.1)–(1.3) is transformed into the new system $\begin{matrix} u_{t t} - ϖ Δ u_{t t} + m Δ^{2} u - γ_{1} Δ θ - γ_{2} Δ P + f (u) = 0, \\ c θ_{t} + d P_{t} + \int_{0}^{\infty} μ (s) B η^{t} (s) d s - γ_{1} Δ u_{t} = 0, \\ d θ_{t} + r P_{t} + \int_{0}^{\infty} λ (s) B ν^{t} (s) d s - γ_{2} Δ u_{t} = 0, \\ η_{t}^{t} + η_{s}^{t} = θ, \\ ν_{t}^{t} + ν_{s}^{t} = P, \end{matrix}$ (2.4) with boundary conditions $u = Δ u = 0, θ = P = 0, η^{t} (\cdot, 0) = ν^{t} (\cdot, 0) = 0 on Γ \times R^{+}$ (2.5) and initial conditions $\begin{matrix} u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), \\ θ (x, 0) = θ_{0} (x), η^{0} (x, t) = η_{0} (x, s), \\ P (x, 0) = P_{0} (x), ν^{0} (x, t) = ν_{0} (x, s) . \end{matrix}$ (2.6)

We shall use the standard Lebesgue spaces and Sobolev spaces with their usual proprieties. Let $⟨ \cdot, \cdot ⟩$ and $∥ \cdot ∥$ denote the $L^{2}$ -inner product and $L^{2}$ -norm, respectively.

Consider the positive operators A and B on $L^{2} (Ω)$ defined by $A = - Δ, B = h I + A,$ with the domain $D (A) = D (B) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ . Now, for $p \in R$ , we introduce the scale of Hilbert spaces $V_{p} = D (A^{p / 2})$ with the usual inner products ${⟨ v_{1}, v_{2} ⟩}_{p} = ⟨ A^{p / 2} v_{1}, A^{p / 2} v_{2} ⟩ .$ We denote by $∥ \cdot ∥_{p}$ the norm on $V_{p}$ induced by the above inner product. The injection $V_{p_{1}} ↪ V_{p_{2}}$ is continuous and compact whenever $p_{1} > p_{2}$ . Denoting by $λ_{1}$ the first eigenvalue of A, for any $s > p$ , we also have the inequalities $∥ A^{p / 2} u ∥ ⩽ λ_{1}^{(p - s) / 2} ∥ A^{s / 2} u ∥ \forall u \in D (A^{s / 2}) .$ (2.7)

We also introduce the biharmonic operator $A_{1} u = Δ^{2} u, u \in D (A_{1}) = {u \in H^{4} (Ω) ∣ u = Δ u = 0 on Γ} .$ (2.8) Let us introduce the inertia operator $M_{ϖ} = I + ϖ A$ , with obvious domain. It is well known that the operator $M_{ϖ}$ is a positive and self-adjoint operator in $V_{0} = L^{2} (Ω)$ . Then, one has $V_{ϖ} : = D (M_{ϖ}^{1 / 2}) \equiv V_{1} = D (A^{1 / 2}) = H_{0}^{1} (Ω) .$

Let $K$ be a memory kernel satisfying the assumptions $(H_{1})$ – $(H_{5})$ . Now for $p \in R$ consider the weighted Hilbert spaces $M_{p, K} = L_{K}^{2} (R^{+}; V_{p}) = {φ : R^{+} \to V_{p} | \int_{0}^{\infty} K (s) {∥ φ (s) ∥}_{V_{p}}^{2} d s},$ equipped with the inner product ${⟨ φ, χ ⟩}_{M_{p, K}} = \int_{0}^{\infty} K (s) ⟨ B^{p / 2} φ (s), B^{p / 2} χ (s) ⟩ d s$ and the norm $∥ φ ∥_{M_{p, K}}^{2} = {⟨ φ, φ ⟩}_{M_{p, K}} .$ We also introduce the linear operator $T$ on $M_{1, K}$ defined by $T φ = - φ_{s}, φ \in D (T),$ (2.9) with the domain $D (T) = {φ \in M_{1, K} ∣ φ_{s} \in M_{1, K}, φ (0) = 0},$ where $φ_{s}$ is the distributional derivative of φ with respect to the internal variable s, and then the operator $T$ is the infinitesimal generator of a $C_{0}$ -semigroup of contractions. Notice that, on account of $(H_{3})$ , there holds ${⟨ T φ, φ ⟩}_{M_{1, K}} = \int_{0}^{\infty} K^{'} (s) {∥ B^{1 / 2} φ (s) ∥}_{2}^{2} d s ⩽ 0 for all φ \in D (T) .$ (2.10) As a direct consequence, we deduce that $\begin{matrix} {⟨ T η, η ⟩}_{M_{1, μ}} = \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}_{2}^{2} d s ⩽ - δ_{1} \int_{0}^{\infty} μ (s) {∥ B^{1 / 2} η (s) ∥}_{2}^{2} d s = - δ_{1} ∥ η ∥_{M_{1, μ}}, \\ {⟨ T ν, ν ⟩}_{M_{1, λ}} = \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}_{2}^{2} d s ⩽ - δ_{2} \int_{0}^{\infty} λ (s) {∥ B^{1 / 2} ν (s) ∥}_{2}^{2} d s = - δ_{2} ∥ ν ∥_{M_{1, λ}} . \end{matrix}$ (2.11) Finally, we define the operator $L_{K} : D (L) \to V_{0}$ $L_{K} φ = \int_{0}^{\infty} K (s) B φ (s) d s$ (2.12) with the domain $D (L_{K}) = {φ \in M_{1, K} | \int_{0}^{\infty} K (s) B φ (s) d s \in V_{0}, φ (0) = 0} .$

Remark 2.1.

Note that the assumptions on the memory kernels μ and λ can be weakened. For instance, they can be unbounded in a neighborhood of the origin (see e.g. [12]).

Actually, even the crucial condition $(H_{4})$ can be relaxed. However, this would be a more delicate task. In fact, the results of this paper hold even if the memory kernels μ and λ satisfy for some $C ⩾ 1$ the weaker conditions [12] $μ (t + s) ⩽ C e^{- δ_{1} t} μ (s) and λ (t + s) ⩽ C e^{- δ_{2} t} λ (s)$ for every $t ⩾ 0$ and almost every $s \in R^{+}$ , provided that μ and λ are not too flat (cf. [24]).

For every $ϖ > 0$ , we introduce the following Hilbert spaces $V_{ϖ} = V_{2} \times V_{ϖ} \times V_{0} \times M_{1, μ} \times V_{0} \times M_{1, λ} .$ Concerning the nonlinear term f, we assume that $f \in C^{1} (R),$ (2.13) and there exist $ς ⩾ 1$ and $C_{f} > 0$ such that $| f^{'} (s) | ⩽ C_{f} (1 + | s |^{ς - 1}) \forall s \in R .$ (2.14) We also assume there exist a constant $c_{f} > 0$ such that $F (s) ⩾ - c_{f} \forall s \in R,$ (2.15) where $F (s) = \int_{0}^{s} f (τ) d τ$ .

From now and throughout the paper, for the sake of simplicity, we write $η = η^{t} (s)$ and $ν = ν^{t} (s)$ . We assume that all the physical parameters of system (2.4) are strictly positive and that conditions (1.4), $(H_{1})$ – $(H_{5})$ and (2.13)–(2.15) hold. Without loss of generality, we shall restrict our attention to the case $0 < ϖ ⩽ 1$ .

Introducing the new variable $v = u_{t}$ , setting $U (t) = (u (t), v (t), θ (t), η, P (t), ν)$ , problem (2.4)–(2.6) can be written as a linear evolution equation in $V_{ϖ}$ of the form $\begin{matrix} \frac{d U (t)}{d t} = A U (t) + F (U (t)), \\ U (0) = U_{0}, \end{matrix}$ (2.16) where $U (0) = (u_{0}, v_{0}, θ_{0}, η_{0}, P_{0}, ν_{0}) \in V_{ϖ}$ , $F (U) = {(0, - M_{ϖ}^{- 1} f (u), 0, 0, 0, 0)}^{T}$ and $A : D (A) \subset V_{ϖ} \to V_{ϖ}$ is the linear operator defined by $A (\begin{matrix} u \\ v \\ θ \\ η \\ P \\ ν \end{matrix}) = (\begin{matrix} v \\ - M_{ϖ}^{- 1} A (m A u - γ_{1} θ - γ_{2} P) \\ \frac{d γ_{2} - r γ_{1}}{c r - d^{2}} A v - \frac{r}{c r - d^{2}} \int_{0}^{\infty} μ (s) B η (s) d s + \frac{d}{c r - d^{2}} \int_{0}^{\infty} λ (s) B ν^{t} (s) d s \\ θ + T η \\ \frac{d γ_{1} - c γ_{2}}{c r - d^{2}} A v + \frac{d}{c r - d^{2}} \int_{0}^{\infty} μ (s) B η (s) d s - \frac{c}{c r - d^{2}} \int_{0}^{\infty} λ (s) B ν^{t} (s) d s \\ P + T ν \end{matrix})$ (2.17) with the domain $D (A) = \{U \in V_{ϖ} |\begin{matrix} m A u - γ_{1} θ - γ_{2} P \in V_{2} \\ v \in V_{2} \\ θ, P \in V_{1} \\ L_{μ} η, L_{λ} ν \in V_{0} \\ η, ν \in D (T) \end{matrix}\} .$

Now, we use the theory of semigroups of linear operators to obtain the existence of solutions to the system (2.4)–(2.6).

Lemma 2.1.

The operator $A$ satisfies the inequality $⟨ A U, U ⟩ ⩽ 0$ for every $U \in D (A)$ .

Proof.

Let $U (t) = (u (t), v (t), θ (t), η, P (t), ν)$ be in $D (A)$ . From (2.17), the divergence theorem and the boundary conditions, it is quite easy to check that $\begin{array}{rclr} {⟨ A U, U ⟩}_{V_{ϖ}} & = & {⟨ T η, η ⟩}_{M_{1, μ}} + {⟨ T ν, ν ⟩}_{M_{1, λ}} \\ = & \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s + \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s ⩽ 0, & (2.18) \end{array}$ where the last inequality is a consequence of (2.10). This proves that $A$ is a dissipative operator. □

Lemma 2.2.

The operator $A$ has the property that $Range (I - A) = V_{ϖ},$ where $I$ is the identity operator in $V_{ϖ}$ .

Proof.

Let $U^{*} = (u^{*}, v^{*}, θ^{*}, η^{*}, P^{*}, ν^{*}) \in V_{ϖ}$ . We must prove that $U - A U = U^{*}$ (2.19) has a solution $U = (u, v, θ, η, P, ν)$ in $D (A)$ . This equation leads to the system $\begin{matrix} u^{*} = u - v, \\ v^{*} = v + M_{ϖ}^{- 1} A (m A u - γ_{1} θ - γ_{2} P), \\ θ^{*} = θ - \frac{d γ_{2} - r γ_{1}}{c r - d^{2}} A v + \frac{r}{c r - d^{2}} \int_{0}^{\infty} μ (s) B η (s) d s - \frac{d}{c r - d^{2}} \int_{0}^{\infty} λ (s) B ν (s) d s, \\ η^{*} = η - θ - T η, \\ P^{*} = P - \frac{d γ_{1} - c γ_{2}}{c r - d^{2}} A v - \frac{d}{c r - d^{2}} \int_{0}^{\infty} μ (s) B η (s) d s + \frac{c}{c r - d^{2}} \int_{0}^{\infty} λ (s) B ν (s) d s, \\ ν^{*} = ν - P - T ν . \end{matrix}$ (2.20) Integrating (2.20)₄ and (2.20)₆ with $η (0) = ν (0) = 0$ leads to $η (s) = (1 - e^{- s}) θ + \int_{0}^{s} e^{τ - s} η^{*} (τ) d τ, ν (s) = (1 - e^{- s}) P + \int_{0}^{s} e^{τ - s} ν^{*} (τ) d τ .$ (2.21) Substituting (2.21)₁ into (2.20)₃ and (2.21)₂ into (2.20)₅, we obtain the following system $\begin{matrix} u + M_{ϖ}^{- 1} A (m A u - γ_{1} θ - γ_{2} P) = u^{*} + v^{*}, \\ (c r - d^{2}) θ + r c_{μ} B θ - d c_{λ} B P - (d γ_{2} - r γ_{1}) A u = Ψ_{1}, \\ (c r - d^{2}) P + d c_{μ} B θ - c c_{λ} B P + (d γ_{1} - c γ_{2}) A u = Ψ_{2}, \end{matrix}$ (2.22) where $\begin{array}{rcl} Ψ_{1} = (c r - d^{2}) θ^{*} - (d γ_{2} - r γ_{1}) A u^{*} - r \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} B η^{*} (τ) d τ d s \\ + d \int_{0}^{\infty} λ (s) \int_{0}^{s} e^{τ - s} B ν^{*} (τ) d τ d s, \\ Ψ_{2} = (c r - d^{2}) P^{*} - (d γ_{1} - c γ_{2}) A u^{*} + d \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} B η^{*} (τ) d τ d s \\ - c \int_{0}^{\infty} λ (s) \int_{0}^{s} e^{τ - s} B ν^{*} (τ) d τ d s, \end{array}$ and $c_{μ} = \int_{0}^{\infty} μ (s) (1 - e^{- s}) d s, c_{λ} = \int_{0}^{\infty} λ (s) (1 - e^{- s}) d s$ (2.23) are positive constants in force of $(H_{1})$ – $(H_{2})$ . It is not hard to check that the right-hand side of (2.22)₁ belongs to $V_{0}$ . Indeed, since μ and λ are decreasing according to $(H_{3})$ , it is quite easy to check that (see [15] for a similar calculation) ${∥ \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} A η^{*} (τ) d τ d s ∥}_{1} ⩽ \int_{0}^{\infty} μ (τ) ∥ η^{*} (τ) ∥ d τ < \infty,$ and $\begin{matrix} {∥ \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} η^{*} (τ) d τ d s ∥}_{1} & ⩽ \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} {∥ η^{*} (τ) ∥}_{1} d τ d s \\ ⩽ \int_{0}^{\infty} \sqrt{μ (s)} \int_{0}^{s} e^{τ - s} \sqrt{μ (τ)} {∥ η^{*} (τ) ∥}_{1} d τ d s \\ ⩽ \sqrt{μ_{0}} {∥ η^{*} (τ) ∥}_{M_{1, μ}} < \infty . \end{matrix}$ Combining the two last inequalities, we conclude that $∥ \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} B η^{*} (τ) d τ d s ∥ < \infty,$ (2.24) which readily yields $Ψ_{1} \in V_{- 1}$ . Similarly, we get $Ψ_{2} \in V_{- 1}$ .

We associate to (2.22) the following bilinear form on $V_{2} \times V_{0} \times V_{0}$ $\begin{array}{rcl} M ((u, θ, P), (\tilde{u}, \tilde{θ}, \tilde{P})) \\ = (u, \tilde{u}) - ϖ {(u, \tilde{u})}_{1} + m {(u, \tilde{u})}_{2} - γ_{1} {(θ, \tilde{u})}_{1} - γ_{2} {(P, \tilde{u})}_{1} + (c r - d^{2}) (θ, \tilde{θ}) \\ + r c_{μ} {(θ, \tilde{θ})}_{1} - d c_{λ} {(P, \tilde{θ})}_{1} - (d γ_{2} - r γ_{1}) {(u, \tilde{θ})}_{1} + (c r - d^{2}) (P, \tilde{P}) + d c_{μ} {(θ, \tilde{P})}_{1} \\ - c c_{λ} {(P, \tilde{P})}_{1} + (d γ_{1} - c γ_{2}) {(u, \tilde{P})}_{1} . \end{array}$

Clearly, $M$ is coercive on $V_{2} \times V_{0} \times V_{0}$ . Moreover, we have $\begin{array}{rcl} | M ((u, θ, P), (\tilde{u}, \tilde{θ}, \tilde{P})) | & ⩽ & C [∥ u ∥_{2} ∥ \tilde{u} ∥_{2} + ∥ θ ∥_{1} ∥ \tilde{θ} ∥_{1} + ∥ P ∥_{1} ∥ \tilde{P} ∥_{1} + ∥ \tilde{u} ∥_{1} ∥ θ ∥_{1} + ∥ u ∥_{1} ∥ \tilde{θ} ∥_{1} \\ + ∥ \tilde{u} ∥_{1} ∥ P ∥_{1} + ∥ u ∥_{1} ∥ \tilde{P} ∥_{1} + ∥ \tilde{θ} ∥_{1} ∥ P ∥_{1} + ∥ θ ∥_{1} ∥ \tilde{P} ∥_{1}] \\ ⩽ & C {∥ (u, θ, P) ∥}_{V_{2} \times V_{0} \times V_{0}} {∥ (\tilde{u}, \tilde{θ}, \tilde{P}) ∥}_{V_{2} \times V_{0} \times V_{0}} \end{array}$ for some constant $C ⩾ 0$ . Hence, by means of the Lax–Milgram theorem, the elliptic problem (2.22) admits a unique (weak) solution $(u, θ, P) \in V_{2} \times V_{1} \times V_{1}$ . Moreover, in light of (2.21), we have $\begin{matrix} ∥ η ∥_{M_{1, μ}}^{2} & ⩽ 2 μ_{0} ∥ θ ∥_{1}^{2} + 2 \int_{0}^{\infty} μ (s) {∥ \int_{0}^{s} e^{τ - s} η^{*} (τ) d τ ∥}_{1}^{2} d s \\ ⩽ 2 μ_{0} ∥ θ ∥_{1}^{2} + 2 \int_{0}^{\infty} {(\int_{0}^{s} e^{τ - s} \sqrt{μ (τ)} {∥ η^{*} (τ) ∥}_{1} d τ)}^{2} d s \\ ⩽ 2 μ_{0} ∥ θ ∥_{1}^{2} + 2 {∥ η^{*} ∥}_{M_{1, μ}}^{2} . \end{matrix}$ (2.25)

Analogously, we obtain $∥ ν ∥_{M_{1, λ}}^{2} ⩽ 2 λ_{0} ∥ P ∥_{1}^{2} + 2 {∥ ν^{*} ∥}_{M_{1, λ}}^{2},$ implying $η \in M_{1, μ}$ and $ν \in M_{1, λ}$ . Accordingly, $T η = η - θ - η^{*} \in M_{1, μ}, T ν = ν - P - ν^{*} \in M_{1, λ} .$ Finally, it is easy to ascertain that $η (s), ν (s) \to 0$ in $V_{1}$ as $s \to 0$ . Thus, in light of the above results, the vector $U = (u, v, θ, η, P, ν) \in D (A)$ , with $v = u - u^{*}$ , $η (s) = η (s; θ)$ and $ν (s) = ν (s; P)$ , solves Eq. (2.19). This finishes to proof. □

We thus have the following theorem.

Theorem 2.1.

The operator $A$ generates a semigroup of contractions in $V_{ϖ}$ .

Proof.

Since the operator $A$ is maximal dissipative in $V_{ϖ}$ and $D (A)$ is densely defined in $V_{ϖ}$ , the proof follows from the Lumer–Phillips corollary to the Hille–Yosida theorem [25]. □

It is worth remarking that this theorem implies that the dynamical system generated by the equations of thermoelasticity with diffusion and memory is stable in the sense of Lyapunov.

We are now in a position to give the definitions of a mild solution (according to [25], Chapter 6) and of a weak solution to problem (2.4)–(2.6).

Definition 2.1.

Consider any $T > 0$ . A solution $U (t) \in C ([0, T]; V_{ϖ})$ to the integral equation $U (t) = e^{A t} U (0) + \int_{0}^{t} e^{A (t - s)} F (U (s)) d s$ (2.26) is called a mild solution to (2.4)–(2.6) on the interval $[0, T]$ .

We say that a set of functions $(u, u_{t}, θ, η, P, ν)$ is a weak solution to (2.4)–(2.6) if $u \in L^{\infty} ([0, T]; V_{2}), u_{t} \in L^{\infty} ([0, T]; V_{ϖ}), θ \in L^{\infty} ([0, T]; V_{0}),$ $η \in L^{\infty} ([0, T]; M_{1, μ}), P \in L^{\infty} ([0, T]; V_{0}), ν \in L^{\infty} ([0, T]; M_{1, λ}),$ (2.6) is satisfied and $\begin{matrix} \int_{0}^{T} \int_{Ω} (- u_{t} ϕ_{1, t} + ϖ u_{t} Δ ϕ_{1, t} + m Δ u Δ ϕ_{1} + γ_{1} θ Δ ϕ_{1} + γ_{2} P Δ ϕ_{1} + f (u) ϕ_{1}) d x d t \\ + \int_{0}^{T} \int_{Ω} (- c θ - d P + γ_{1} Δ u) ϕ_{2, t} d x d t - \int_{0}^{T} \int_{Ω} (\int_{0}^{\infty} μ (s) Δ η (s) d s) ϕ_{2} d x d t \\ + \int_{0}^{T} \int_{Ω} (- d θ - r P + γ_{2} Δ u) ϕ_{3, t} d x d t - \int_{0}^{T} \int_{Ω} (\int_{0}^{\infty} λ (s) Δ ν (s) d s) ϕ_{3} d x d t \\ + \int_{0}^{T} \int_{Ω} (- η ϕ_{4, t} + η_{s} ϕ_{4} - θ ϕ_{4}) d x d t + \int_{0}^{T} \int_{Ω} (- ν ϕ_{5, t} + ν_{s} ϕ_{5} - P ϕ_{5}) d x d t \\ - \int_{Ω} (u_{1} ϕ_{1} (0) - ϖ u_{1} Δ ϕ_{1} (0) + (c θ_{0} + d P_{0} - γ_{1} Δ u_{0}) ϕ_{2} (0) \\ + (d θ_{0} + r P_{0} - γ_{2} Δ u_{0}) ϕ_{3} (0) + η_{0} ϕ_{4} (0) + ν_{0} ϕ_{5} (0)) d x = 0 \end{matrix}$ (2.27) for all $ϕ_{1} \in C^{1} ([0, T]; V_{2})$ , $(ϕ_{2}, ϕ_{3}) \in C^{1} ([0, T]; V_{0} \times V_{0})$ , $ϕ_{4} \in C^{1} ([0, T]; M_{1, μ})$ and $ϕ_{5} \in C^{1} ([0, T]; M_{1, λ})$ such that $ϕ_{1} (T) = ϕ_{2} (T) = ϕ_{3} (T) = ϕ_{4} (T) = ϕ_{5} (T) = 0 .$

Now an application of the theory of semigroups [25] gives the following theorem. Theorem 2.2.

Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold.

Then for every initial condition $U_{0} = (u_{0}, u_{1}, θ_{0}, η_{0}, P_{0}, ν_{0}) \in V_{ϖ}$ , problem ( 2.4 )–( 2.6 ) has a unique global mild solution satisfying $U (t) \in C ([0, \infty); V_{ϖ})$ .

If $U_{0} = (u_{0}, u_{1}, θ_{0}, η_{0}, P_{0}, ν_{0}) \in D (A)$ , then the corresponding mild solution is strong, that is, it is continuously differentiable, it takes values in $D (A)$ and it satisfies ( 2.16 ) in $V_{ϖ}$ for almost all $t \in [0, T]$ .

Proof.

We need to prove first that the operator $F (U) = {(0, - M_{ϖ}^{- 1} f (u), 0, 0, 0, 0)}^{T}$ defined in (2.16) is locally Lipschitz in $V_{ϖ}$ . To this end, we recall that $M_{ϖ} : H_{0}^{1} (Ω) \to H^{- 1} (Ω)$ is an isometrical bijection with respect to the norm $∥ u ∥_{H_{0}^{1}}^{2} = ∥ \nabla u ∥_{2}^{2} + ∥ u ∥_{2}^{2}$ and hence ${∥ M_{ϖ}^{- 1} u ∥}_{H_{0}^{1}} = ∥ u ∥_{H^{- 1}} \forall u \in H^{- 1} (Ω) .$

Let us denote $U^{1} (t) = (u^{1} (t), u_{t}^{1} (t), θ^{1} (t), η^{1}, P^{1} (t), ν^{1}) \in V_{ϖ}$ and $U^{2} (t) = (u^{2} (t), u_{t}^{2} (t), θ^{2} (t), η^{2}, P^{2} (t), ν^{2}) \in V_{ϖ}$ . It follows from the mean value theorem and assumption (2.14) that $\begin{array}{rclr} \int (f (u^{1}) - f (u^{2})) χ d x & ⩽ & {∥ f^{'} (ς u^{1} + (1 - ς) u^{2}) ∥}_{2 p}^{p - 1} {∥ u^{1} - u^{2} ∥}_{2 p} ∥ χ ∥_{2} \\ ⩽ & C_{f}^{'} {∥ \nabla u^{1} - \nabla u^{2} ∥}_{2} ∥ χ ∥_{2}, & (2.28) \end{array}$ where $0 ⩽ ς ⩽ 1$ and $C_{f}^{'} > 0$ is a constant depending on the initial data. Therefore, we have ${∥ f (u^{1}) - f (u^{2}) ∥}_{2} ⩽ C_{f}^{'} {∥ \nabla u^{1} - \nabla u^{2} ∥}_{2}$ and consequently $\begin{matrix} {∥ F (U^{1}) - F (U^{2}) ∥}_{V_{ϖ}} & ⩽ {∥ M_{ϖ}^{- 1} (f (u^{1}) - f (u^{2})) ∥}_{H_{0}^{1}} \\ ⩽ {∥ f (u^{1}) - f (u^{2}) ∥}_{H^{- 1}} \\ ⩽ C_{f}^{'} {∥ \nabla u^{1} - \nabla u^{2} ∥}_{2} \\ ⩽ C {∥ U^{1} - U^{2} ∥}_{V_{ϖ}} . \end{matrix}$ (2.29)

It follows that $F$ is locally Lipschitz in $V_{ϖ}$ . Combining this with Theorem 2.1, we have from a classical result (see [25], Theorem 6.1.4) that the Cauchy problem (2.16) has a unique local mild solution given by (2.26) and defined in a maximal interval $(0, t_{max})$ .

Next we prove that the solution is global, that is, $t_{max} = \infty$ .

Taking the inner product of (2.16) and $U (t) \in V_{ϖ}$ , we get from (2.18) $\frac{d}{d t} E (t) = \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s + \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s,$ (2.30) where $\begin{array}{rclr} E (t) & = & \frac{1}{2} {∥ U (t) ∥}_{V_{ϖ}}^{2} + \int_{Ω} F (u) d x \\ = & \frac{1}{2} (∥ u_{t} ∥^{2} + ϖ {∥ A^{1 / 2} u_{t} ∥}^{2} + m ∥ A u ∥^{2} + c ∥ θ ∥^{2} + 2 d ⟨ θ, P ⟩ + r ∥ P ∥^{2} + ∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}) \\ + \int_{Ω} F (u) d x & (2.31) \end{array}$ and $F (u) = \int_{0}^{u} f (τ) d τ$ . The assumption (2.15) implies that $\int_{Ω} F (u) d x ⩾ - c_{f} | Ω | .$ (2.32)

Now, combining (2.31) with (1.5) and (2.32), we get $\begin{array}{rclr} E (t) & = & \frac{1}{2} {∥ U (t) ∥}_{V_{ϖ}}^{2} + \int_{Ω} F (u) d x ⩾ \frac{℘}{2} (∥ A u ∥^{2} + ∥ θ ∥^{2} + ∥ P ∥^{2} + ∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}) - c_{f} | Ω | \\ ⩾ & \frac{℘}{2} {∥ U (t) ∥}_{V_{ϖ}}^{2} - c_{f} | Ω |, & (2.33) \end{array}$ where $℘ ⩾ max {m, c - \frac{d}{κ}, r - d κ, 1} .$ We recall that $κ > 0$ is assumed to be such that $\frac{d}{c} < κ < \frac{r}{d}$ . As a result ${∥ U (t) ∥}_{V_{ϖ}}^{2} ⩽ \frac{2}{℘} (\frac{1}{2} {∥ U (t) ∥}_{V_{ϖ}}^{2} + \int_{Ω} F (u) d x + c_{f} | Ω |) .$ (2.34)

Integrating (2.30) with respect to t, we infer from (2.34) that ${∥ U (t) ∥}_{V_{ϖ}} ⩽ C ({∥ U (0) ∥}_{V_{ϖ}}, | Ω |, f) \forall t ⩾ 0 .$ (2.35)

This uniform estimate together with Theorem 2.5.5 of [25] yields the global existence, i.e., $U (t) \in C (0, \infty; V_{ϖ})$ .

Consider $T > 0$ , any $t \in (0, T)$ and two mild solutions $U^{1} (t)$ and $U^{2} (t)$ with initial data $U^{1} (0)$ and $U^{2} (0)$ , respectively. Let us also assume that $∥ U^{i} ∥_{V_{ϖ}} ⩽ R$ . Then, using (2.26), ${∥ U^{1} (t) - U^{2} (t) ∥}_{V_{ϖ}} ⩽ {∥ e^{t A} (U^{1} (0) - U^{2} (0)) ∥}_{V_{ϖ}} + \int_{0}^{t} {∥ e^{A (t - s)} (F (U^{1} (s)) - F (U^{2} (s))) ∥}_{V_{ϖ}} d s .$ Using the local Lipschitz property of $F$ , (2.35) and $∥ e^{t A} ∥_{[V_{ϖ}, V_{ϖ}]} ⩽ 1$ , we obtain ${∥ U^{1} (t) - U^{2} (t) ∥}_{V_{ϖ}} ⩽ {∥ U^{1} (0) - U^{2} (0) ∥}_{V_{ϖ}} + C_{R, T} \int_{0}^{t} {∥ (U^{1} (s) - U^{2} (s)) ∥}_{V_{ϖ}} d s .$ Then, the Gronwall’s lemma gives ${∥ U^{1} (t) - U^{2} (t) ∥}_{V_{ϖ}}^{2} ⩽ e^{C_{R, T}} {∥ U^{1} (0) - U^{2} (0) ∥}_{V_{ϖ}}^{2}, 0 ⩽ t ⩽ T,$ (2.36) which implies the continuous dependence of the mild solution on the initial data.

The last statement of the theorem (on strong solutions) follows directly from [25] (see Theorem 6.1.5). The proof is complete. □

Remark 2.2.

From the above theorem, we can see that the solution to our problem (2.16) defines a strongly continuous semigroup $S (t)$ on the phase space $V$ such that $S (t) U (0) = U (t)$ .

By using a density argument, one can prove the existence and uniqueness of the weak solution (see [27] for details). In fact, any mild solution is a weak solution.

Theorem 2.2 has been proved in [4] in the case $ϖ = 0$ .

3. Exponential stability of the linear system

We already mentioned that, in absence of the rotational inertia term ( $ϖ = 0$ ), if the thermal memory kernel is not allowed to grow too rapidly around the origin, then the associated semigroup $S (t)$ on $V_{ϖ}$ lacks of exponentially stability. The aim of the present section is to prove that the semigroup $S (t)$ on $V_{ϖ}$ corresponding to the linear system of (2.4)–(2.6), i.e., $f (u) = 0$ , is exponentially stable at a decay rate proportional to ϖ (we take $0 < ϖ ⩽ 1$ for simplicity). Our approach is based on the construction of a Lyapunov functional, namely, we obtain the decay estimate for a suitably defined perturbation of the energy functional $E_{0}$ given by the following result. This result will help us to establish the so-called stabilizability inequality.

We first show an energy estimate that will be used in what follows.

Lemma 3.1.
Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $(u, v, θ, η, P, ν)$ be a solution to the linear problem ( 2.4 )–( 2.6 ) ( $f (u) = 0$ ). For some $δ_{1}$ and $δ_{2} > 0$ satisfying $(H_{4})$ , there holds $\begin{array}{rclr} \frac{d}{d t} E_{0} (t) & = & {⟨ T η, η ⟩}_{M_{1, μ}} + {⟨ T ν, ν ⟩}_{M_{1, λ}} \\ ⩽ & - \frac{δ_{1}}{2} ∥ η ∥_{M_{1, μ}}^{2} - \frac{δ_{2}}{2} ∥ ν ∥_{M_{1, λ}}^{2} + \frac{1}{2} \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s \\ + \frac{1}{2} \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s \\ ⩽ & 0, & (3.1) \end{array}$ where $\begin{array}{rclr} E_{0} (t) & = & \frac{1}{2} (∥ u_{t} ∥^{2} + m ∥ A u ∥^{2} + ϖ {∥ A^{1 / 2} u_{t} ∥}^{2} + c ∥ θ ∥^{2} + 2 d ⟨ θ, P ⟩ \\ + r ∥ P ∥^{2} + ∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}) . & (3.2) \end{array}$
Proof.
Taking the inner product in $V_{ϖ}$ of equation (2.4)₁ (with $f (u) = 0$ ) and $u_{t}$ in $V_{2}$ , (2.4)₂ and θ in $V_{0}$ , (2.4)₃ and P in $V_{0}$ , (2.4)₄ and η in $M_{1, μ}$ and (2.4)₅ and ν in $M_{1, λ}$ , we conclude that $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (∥ u_{t} ∥^{2} + m ∥ A u ∥_{2}^{2} + ϖ {∥ A^{1 / 2} u_{t} ∥}^{2}) = γ_{1} ⟨ A θ, u_{t} ⟩ + γ_{2} ⟨ A P, u_{t} ⟩, \\ \frac{c}{2} \frac{d}{d t} ∥ θ ∥^{2} + d ⟨ P_{t}, θ ⟩ = \int_{0}^{\infty} μ (s) (\int_{Ω} B η (s) θ (t) d x) d s - γ_{1} ⟨ A θ, u_{t} ⟩, \\ \frac{r}{2} \frac{d}{d t} ∥ P ∥^{2} + d ⟨ θ_{t}, P ⟩ = \int_{0}^{\infty} λ (s) (\int_{Ω} B ν (s) P (t) d x) d s - γ_{2} ⟨ A P, u_{t} ⟩, \\ \frac{1}{2} \frac{d}{d t} ∥ η ∥_{M_{1, μ}}^{2} = - \int_{0}^{\infty} μ (s) (\int_{Ω} B η (s) θ (t) d x) d s - {⟨ η_{s}, η ⟩}_{M_{1, μ}} \end{array}$ and $\frac{1}{2} \frac{d}{d t} ∥ ν ∥_{M_{1, λ}}^{2} = - \int_{0}^{\infty} λ (s) (\int_{Ω} B ν (s) P (t) d x) d s - {⟨ ν_{s}, ν ⟩}_{M_{1, λ}} .$ Combining the above equations and taking into account (2.9), we deduce that the first identity of (3.1) holds, while the inequalities follow from (2.10) and (2.11). □
Lemma 3.2.
Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $(u, v, θ, η, P, ν)$ be a solution to the linear problem ( 2.4 )–( 2.6 ) ( $f (u) = 0$ ). Given $σ_{0} > 0$ there exists a positive constant $C_{σ_{0}}$ such that $\begin{array}{rclr} \frac{d}{d t} K (t) & ⩽ & - \frac{γ_{1} + γ_{2}}{2} {∥ u_{t} (t) ∥}^{2} - ϖ (γ_{1} + γ_{2}) {∥ A^{1 / 2} u_{t} (t) ∥}^{2} + \frac{m}{2} σ_{0} {∥ A u (t) ∥}^{2} \\ + ϖ C_{σ_{0}} (∥ θ ∥^{2} + ∥ P ∥^{2}) + ϖ C (∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}), & (3.3) \end{array}$ where $K (t) : = ⟨ u_{t} (t), (ϖ I + A^{- 1}) ((c + d) θ (t) + (r + d) P (t)) ⟩ .$
Proof.
By a direct computation, we get $\begin{array}{rcl} \frac{d}{d t} K (t) & = & (c + d) ϖ ⟨ u_{t t} (t), θ (t) ⟩ + (c + d) ϖ ⟨ u_{t} (t), θ_{t} (t) ⟩ \\ + (c + d) ⟨ u_{t t} (t), A^{- 1} θ (t) ⟩ + (c + d) ⟨ u_{t} (t), A^{- 1} θ_{t} (t) ⟩ \\ + (r + d) ϖ ⟨ u_{t t} (t), P (t) ⟩ + (r + d) ϖ ⟨ u_{t} (t), P_{t} (t) ⟩ \\ + (r + d) ⟨ u_{t t} (t), A^{- 1} P (t) ⟩ + (r + d) ⟨ u_{t} (t), A^{- 1} P_{t} (t) ⟩ . \end{array}$

Taking the inner product of the first equation in (2.4) and $A^{- 1} θ (t)$ , we get $⟨ u_{t t} (t), A^{- 1} θ (t) ⟩ + ϖ ⟨ u_{t t} (t), θ (t) ⟩ + m ⟨ A u (t), θ (t) ⟩ - γ_{1} ∥ θ ∥^{2} - γ_{2} ⟨ P (t), θ (t) ⟩ = 0 .$ The inner product of (2.4)₁ and $A^{- 1} P (t)$ yields $\begin{matrix} ⟨ u_{t t} (t), A^{- 1} P (t) ⟩ + ϖ ⟨ u_{t t} (t), P (t) ⟩ + m ⟨ A u (t), P (t) ⟩ - γ_{2} ∥ P ∥^{2} - γ_{1} ⟨ θ (t), P (t) ⟩ + ⟨ A f (u), P (t) ⟩ \\ = 0 . \end{matrix}$

Furthermore, the inner product of equations (2.4)_2,3 and $ϖ u_{t}$ gives, respectively, $\begin{array}{rcl} c ϖ ⟨ u_{t} (t), θ_{t} (t) ⟩ + d ϖ ⟨ u_{t} (t), P_{t} (t) ⟩ + ϖ γ_{1} {∥ A^{1 / 2} u_{t} (t) ∥}^{2} + ϖ {⟨ u_{t} (t), η^{t} ⟩}_{M_{1, μ}} = 0, \\ d ϖ ⟨ u_{t} (t), θ_{t} (t) ⟩ + r ϖ ⟨ u_{t} (t), P_{t} (t) ⟩ + ϖ γ_{2} {∥ A^{1 / 2} u_{t} (t) ∥}^{2} + ϖ {⟨ u_{t} (t), ν^{t} ⟩}_{M_{1, λ}} = 0 . \end{array}$

The inner product of equations (2.4)_2,3 and $A^{- 1} u_{t}$ , respectively, gives $\begin{array}{rcl} c ⟨ A^{- 1} θ_{t} (t), u_{t} (t) ⟩ + d ⟨ A^{- 1} P_{t} (t), u_{t} (t) ⟩ + {⟨ A^{- 1} u_{t} (t), η^{t} ⟩}_{M_{1, μ}} + γ_{1} {∥ u_{t} (t) ∥}^{2} = 0, \\ d ⟨ A^{- 1} θ_{t} (t), u_{t} (t) ⟩ + r ⟨ A^{- 1} P_{t} (t), u_{t} (t) ⟩ + {⟨ A^{- 1} u_{t} (t), ν^{t} ⟩}_{M_{1, λ}} + γ_{2} {∥ u_{t} (t) ∥}^{2} = 0 . \end{array}$

Hence, by combining the previous identities, we obtain $\begin{array}{rcl} \frac{d}{d t} K (t) & = & - (γ_{1} + γ_{2}) {∥ u_{t} (t) ∥}^{2} + (c + d) γ_{1} ∥ θ ∥^{2} + (r + d) γ_{2} ∥ P ∥^{2} - ϖ (γ_{1} + γ_{2}) {∥ A^{1 / 2} u_{t} (t) ∥}^{2} \\ + γ_{1} (r + d) ⟨ P (t), θ (t) ⟩ + γ_{2} (c + d) ⟨ P (t), θ (t) ⟩ - {⟨ (ϖ I + A^{- 1}) u_{t} (t), η^{t} ⟩}_{M_{1, μ}} \\ - {⟨ (ϖ I + A^{- 1}) u_{t} (t), ν^{t} ⟩}_{M_{1, λ}} - ⟨ m A u (t), (c + d) θ (t) ⟩ - ⟨ m A u (t), (r + d) P (t) ⟩ . \end{array}$ Notice that, by the Young’s inequality, there holds $\begin{array}{rcl} | ⟨ m A u (t), (c + d) θ (t) ⟩ | ⩽ ϖ C_{σ_{0}} {∥ θ (t) ∥}^{2} + \frac{m}{4} σ_{0} {∥ A u (t) ∥}^{2}, \\ | {⟨ ϖ u_{t} (t), η^{t} ⟩}_{M_{1, μ}} | ⩽ \frac{γ_{1} + γ_{2}}{8} {∥ u_{t} (t) ∥}^{2} + ϖ C ∥ η ∥_{M_{1, μ}}^{2}, \\ | {⟨ A^{- 1} u_{t} (t), η^{t} ⟩}_{M_{1, μ}} | ⩽ \frac{γ_{1} + γ_{2}}{8} {∥ u_{t} (t) ∥}^{2} + ϖ C ∥ η ∥_{M_{1, μ}}^{2} . \end{array}$ Combining the previous inequalities with those corresponding to P and ν, we arrive at (3.3). □
Lemma 3.3.
Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $(u, v, θ, η, P, ν)$ be a solution to the linear problem ( 2.4 )–( 2.6 ). Then there holds $\frac{d}{d t} R (t) ⩽ {∥ u_{t} (t) ∥}^{2} + ϖ {∥ A^{1 / 2} u_{t} (t) ∥}^{2} - \frac{m}{2} {∥ A u (t) ∥}^{2} + C^{'} ({∥ θ (t) ∥}^{2} + {∥ P (t) ∥}^{2}),$ (3.4) where $R (t) : = ⟨ u_{t} (t), u (t) ⟩ + ϖ ⟨ A^{1 / 2} u_{t} (t), A^{1 / 2} u (t) ⟩ .$
Proof.
By direct computations, we get $\begin{array}{rcl} \frac{d}{d t} R (t) & = & ⟨ u_{t t} (t), u (t) ⟩ + {∥ u_{t} (t) ∥}^{2} + ϖ ⟨ A^{1 / 2} u_{t t} (t), A^{1 / 2} u (t) ⟩ + ϖ {∥ A^{1 / 2} u_{t} (t) ∥}^{2} \\ = & {∥ u_{t} (t) ∥}^{2} + ϖ {∥ A^{1 / 2} u_{t} (t) ∥}^{2} - m {∥ A u (t) ∥}^{2} + γ_{1} ⟨ A θ (t), u (t) ⟩ + γ_{2} ⟨ A P (t), u (t) ⟩ . \end{array}$

Using the above estimates we get the desired result. □
Lemma 3.4.
Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $U (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν)$ be a solution to the linear problem ( 2.4 )–( 2.6 ). Given $σ_{1} > 0$ there exists two constants $C_{σ_{1}} > 0$ and $C_{σ_{1}}^{'} > 0$ such that $\begin{array}{rclr} \frac{d}{d t} (Z_{1} (t) + Z_{2} (t)) & ⩽ & - \hat{c} {∥ θ (t) ∥}^{2} - \hat{r} {∥ P (t) ∥}^{2} + C_{σ_{1}} ∥ η ∥_{M_{1, μ}}^{2} + C_{σ_{1}}^{'} ∥ ν ∥_{M_{1, λ}}^{2} \\ - c_{μ} \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s - c_{λ} \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s \\ + 2 σ_{1} {∥ A^{1 / 2} u_{t} (t) ∥}^{2}, & (3.5) \end{array}$ where $\begin{matrix} Z_{1} (t) : = - \int_{0}^{\infty} μ (s) ⟨ c θ (t) + d P (t), η ⟩ d s, Z_{2} (t) : = - \int_{0}^{\infty} λ (s) ⟨ d θ (t) + r P (t), ν ⟩ d s, \\ \hat{c} = \frac{1}{2} (μ_{0} c - (μ_{0} + λ_{0}) \frac{d}{κ}) > 0, \hat{r} = \frac{1}{2} (λ_{0} r - (μ_{0} + λ_{0}) d κ) > 0, \end{matrix}$ (3.6) and where $κ > 0$ is assumed to be such that $\frac{d}{c} < κ < \frac{r}{d}$ .
Proof.
By a direct computation, we get $\frac{d}{d t} Z_{1} (t) = - \int_{0}^{\infty} μ (s) ⟨ c θ_{t} (t) + d P_{t} (t), η ⟩ d s - \int_{0}^{\infty} μ (s) ⟨ c θ (t) + d P (t), η_{t} ⟩ d s .$ From (2.4)₄ we get $\begin{array}{rclr} \frac{d}{d t} Z_{1} (t) & = & - \int_{0}^{\infty} μ (s) ⟨ c θ_{t} (t) + d P_{t} (t), η ⟩ d s + c \int_{0}^{\infty} μ (s) ⟨ θ (t), η_{s} ⟩ d s \\ - c μ_{0} {∥ θ (t) ∥}^{2} + d \int_{0}^{\infty} μ (s) ⟨ P (t), η_{s} ⟩ d s - d \int_{0}^{\infty} μ (s) ⟨ θ (t), P (t) ⟩ d s . & (3.7) \end{array}$

The first term on the right-hand side of (3.7) is given by $\begin{matrix} - \int_{0}^{\infty} μ (s) ⟨ c θ_{t} (t) + d P_{t} (t), η ⟩ d s = & γ_{1} \int_{0}^{\infty} μ (s) ⟨ A u_{t} (t), η ⟩ d s \\ + \int_{Ω} (\int_{0}^{\infty} μ (s) B η (s) d s) (\int_{0}^{\infty} μ (s) η (s) d s) d x, \end{matrix}$ and can be controlled in the following way $\begin{array}{rcl} | γ_{1} \int_{0}^{\infty} μ (s) ⟨ A u_{t} (t), η ⟩ d s | ⩽ σ_{1} {∥ A u_{t} (t) ∥}^{2} + C (σ_{1}) ∥ η ∥_{M_{1, μ}}^{2}, \\ | \int_{Ω} (\int_{0}^{\infty} μ (s) B η (s) d s) (\int_{0}^{\infty} μ (s) η (s) d s) d x | ⩽ μ_{0} ∥ η ∥_{M_{1, μ}}^{2} . \end{array}$ Moreover, by integration by parts, we get $\begin{array}{rclr} | c \int_{0}^{\infty} μ (s) ⟨ θ (t), η_{s} ⟩ d s | & = & c | - \int_{0}^{\infty} μ^{'} (s) ⟨ θ (t), η ⟩ d s | \\ ⩽ & - c \int_{0}^{\infty} μ^{'} (s) {∥ θ (t) ∥}^{2} ∥ η ∥^{2} d s \\ ⩽ & \frac{μ_{0} c}{4} {∥ θ (t) ∥}^{2} - c_{μ_{0}} \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s . & (3.8) \end{array}$ Analogously, we obtain $| d \int_{0}^{\infty} μ (s) ⟨ P (t), η_{s} ⟩ d s | ⩽ \frac{λ_{0} r}{4} {∥ P (t) ∥}^{2} - c_{μ_{0}}^{'} \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s,$ (3.9) where $c_{μ_{0}} > 0$ and $c_{μ_{0}}^{'} > 0$ . Using (1.5) we get $| - d \int_{0}^{\infty} μ (s) ⟨ θ (t), P (t) ⟩ d s | ⩽ μ_{0} \frac{d}{2 κ} {∥ θ (t) ∥}^{2} + μ_{0} \frac{d κ}{2} {∥ P (t) ∥}^{2} .$ (3.10)

Then we obtain $\begin{array}{rclr} \frac{d}{d t} Z_{1} (t) & ⩽ & \frac{μ_{0}}{2} (- \frac{3 c}{4} + \frac{d}{κ}) {∥ θ (t) ∥}^{2} + \frac{1}{2} (μ_{0} d κ + \frac{λ_{0} r}{2}) {∥ P (t) ∥}^{2} - C_{σ_{1}} \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s \\ + c_{μ} ∥ η ∥_{M_{1, μ}}^{2} + σ_{1} {∥ A^{1 / 2} u_{t} (t) ∥}^{2}, & (3.11) \end{array}$ where $C_{σ_{1}} = c_{μ_{0}} + c_{μ_{0}}^{'}$ , $c_{μ} = μ_{0} + C (σ_{1}) > 0$ . Then using the same arguments, we can obtain $\begin{array}{rclr} \frac{d}{d t} Z_{2} (t) & ⩽ & \frac{1}{2} (λ_{0} \frac{d}{κ} + \frac{μ_{0} c}{2}) {∥ θ (t) ∥}^{2} + \frac{λ_{0}}{2} (- \frac{3 r}{2} + d κ) {∥ P (t) ∥}^{2} - C_{σ_{1}}^{'} \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ζ (s) ∥}^{2} d s \\ + c_{λ} ∥ ν ∥_{M_{1, λ}}^{2} + σ_{1} {∥ A^{1 / 2} u_{t} (t) ∥}^{2}, & (3.12) \end{array}$ where $c_{λ}, C_{σ_{1}}^{'} > 0$ . Adding (3.11) and (3.12) we arrive at (3.5).

We choose κ in such a way that we get $\hat{c} = \frac{1}{2} (μ_{0} c - (μ_{0} + λ_{0}) \frac{d}{κ}) > 0 and \hat{r} = \frac{1}{2} (λ_{0} r - (μ_{0} + λ_{0}) d κ) > 0,$ (3.13) which implies $\frac{d}{c} < \frac{μ_{0} + λ_{0}}{μ_{0}} \frac{d}{c} < κ < \frac{λ_{0}}{μ_{0} + λ_{0}} \frac{r}{d} < \frac{r}{d} .$ Then the estimate (3.5) holds with (3.6). □

The main result of this section is the following theorem. Theorem 3.1.
Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $(u, v, θ, η, P, ν)$ be a solution to the linear problem. Then, there exist two positive constants M and $c_{0}$ , both independent of ϖ, such that $E_{0} (t) ⩽ M E_{0} (0) e^{- c_{0} ϖ t} \forall t ⩾ 0 .$ (3.14)
Proof.
Let us define a Lyapunov functional by $ℵ (t) = ϖ N E_{0} (t) + ε_{0} K (t) + ϖ ε_{1} R (t) + ϖ ε_{2} (Z_{1} (t) + Z_{2} (t)) .$ We easily see that, for N enough large, there exist $C_{1}, C_{2} > 0$ , such that $C_{1} E_{0} (t) ⩽ ℵ (t) ⩽ C_{2} E_{0} (t) .$ (3.15)

On account of the formulas obtained in the previous lemmas of this section, we deduce that $\begin{array}{rclr} \frac{d}{d t} ℵ (t) & ⩽ & - (\frac{γ_{1} + γ_{2}}{2} ε_{0} - ϖ ε_{1}) {∥ u_{t} (t) ∥}^{2} - ϖ ((γ_{1} + γ_{2}) ε_{0} - ϖ ε_{1} - 2 σ_{1} ε_{2}) {∥ A^{1 / 2} u_{t} (t) ∥}^{2} \\ + \frac{m}{2} (σ_{0} ε_{0} - ϖ ε_{1}) {∥ A u (t) ∥}^{2} + ϖ (C_{σ_{0}} ε_{0} + C^{'} ε_{1} - \hat{c} ε_{2}) ∥ θ ∥^{2} \\ + ϖ (C_{σ_{0}} ε_{0} + C^{'} ε_{1} - \hat{r} ε_{2}) ∥ P ∥^{2} + ϖ (C ε_{0} + C_{σ_{1}} ε_{2} - \frac{δ_{1}}{2} N) ∥ η ∥_{M_{1, μ}}^{2} \\ + ϖ (C ε_{0} + C_{σ_{1}}^{'} ε_{2} - \frac{δ_{2}}{2} N) ∥ ν ∥_{M_{1, λ}}^{2} + ϖ (\frac{N}{2} - c_{μ} ε_{2}) \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s \\ + ϖ (\frac{N}{2} - c_{λ} ε_{2}) \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s . & (3.16) \end{array}$

Choosing now $σ_{0} = \frac{ϖ ε_{1}}{2 ε_{0}}, σ_{1} = \frac{γ_{1} + γ_{2}}{4 ε_{2}} ε_{0}, ε_{0} = \frac{3 ϖ ε_{1}}{γ_{1} + γ_{2}},$ we get $\begin{array}{rcl} \frac{d}{d t} ℵ (t) & ⩽ & - \frac{ϖ ε_{1}}{2} {∥ u_{t} (t) ∥}^{2} - \frac{ϖ^{2} ε_{1}}{2} {∥ A^{1 / 2} u_{t} (t) ∥}^{2} - \frac{m ϖ ε_{1}}{4} {∥ A u (t) ∥}^{2} \\ + ϖ (C_{σ_{0}} ε_{0} + C^{'} ε_{1} - \hat{c} ε_{2}) ∥ θ ∥^{2} + ϖ (C_{σ_{0}} ε_{0} + C^{'} ε_{1} - \hat{r} ε_{2}) ∥ P ∥^{2} \\ + ϖ (C ε_{0} + C_{σ_{1}} ε_{2} - \frac{δ_{1}}{2} N) ∥ η ∥_{M_{1, μ}}^{2} + ϖ (C ε_{0} + C_{σ_{1}}^{'} ε_{2} - \frac{δ_{2}}{2} N) ∥ ν ∥_{M_{1, λ}}^{2} \\ + ϖ (\frac{N}{2} - c_{μ} ε_{2}) \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s \\ + ϖ (\frac{N}{2} - c_{λ} ε_{2}) \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s . \end{array}$

Choosing then $ε_{2} > max {\frac{C ε_{0} + C^{'} ε_{1}}{\hat{c}}; \frac{C ε_{0} + C^{'} ε_{1}}{\hat{r}}}$ and $N > max {\frac{2}{δ_{1}} (C ε_{0} + C_{σ_{1}} ε_{2}); \frac{2}{δ_{2}} (C ε_{0} + C_{σ_{1}} ε_{2}); 2 ε_{2} c_{μ}; 2 ε_{2} c_{λ}},$ we get $\frac{d}{d t} ℵ (t) ⩽ - ϖ C E_{0} (t)$ for some positive constant C which is independent of ϖ. Using (3.15), this proves our theorem. □
Remark 3.1.
The energy exponential stability decay rate which appears in Theorem 3.1 breaks down in the case $ϖ = 0$ . The proof of this lack of exponential stability is omitted here for the sake of brevity (see e.g. [17] and [19] for the case of thermoelastic plates with memory).

4. Global attractors

In this section we prove the existence of a compact global attractor of the dynamical system $(V_{ϖ}, S (t))$ , its finite fractal dimension and some of its properties.

For the reader’s convenience, we present here some definitions related to global attractors for gradient systems that can be found in classical works (see e.g. [21,22] and [28]) or more recent references such as [23] and [7].

Let $S (t)$ be a strongly continuous semigroup and $(V_{ϖ}, S (t))$ be a dynamical system given by $S (t)$ on $V_{ϖ}$ . We give the following definitions.

Definition 4.1.

We say that $(V_{ϖ}, S (t))$ is asymptotically smooth if for any bounded positively invariant set $B \subset V_{ϖ}$ , there exists a compact set $K \subset \overline{B}$ such that $lim_{t \to \infty} {dist}_{V_{ϖ}} (S (t) B, K) = 0,$ where ${dist}_{V_{ϖ}}$ is the Hausdorff semi-distance in $V_{ϖ}$ .

A global attractor $S (t)$ is a compact set $A$ of $V_{ϖ}$ that is fully invariant and attracting, that is, $S (t) A = A$ for all $t ⩾ 0$ and attracts any bounded set $B$ of $V_{ϖ}$ in terms of the Hausdorff semi-distance, i.e., ${dist}_{V_{ϖ}} (S (t) B, A) = sup_{x \in B} inf_{y \in A} {∥ S (t) x - y ∥}_{V_{ϖ}} \to 0 as t \to \infty .$

Let $N$ be the set of steady states of $S (t)$ , $N = {U \in V_{ϖ} ∣ S (t) U = U for all t ⩾ 0} .$ (4.1)

We define the unstable manifold $M^{u} (N)$ emanating from $N$ as the set of all $U \in V_{ϖ}$ such that there exists a full trajectory $U (0) = U and lim_{t \to - \infty} {dist}_{V_{ϖ}} (U (t), N) = 0 .$

It follows directly from the definition that a global attractor for $(V_{ϖ}, S (t))$ is a collection of all bounded full trajectories of the semigroup $S (t)$ and $M^{u} (N)$ is an invariant set. It is also easy to prove that if the dynamical system $(V_{ϖ}, S (t))$ possesses a global attractor $A$ , then $M^{u} (N) \subset A$ . For gradient systems it is possible to prove that $M^{u} (N) = A$ .

We give the following definition of gradient systems.

Definition 4.2.

We call a system $(V_{ϖ}, S (t))$ a gradient system if it possesses a strict Lyapunov functional, that is, a continuous functional $Φ : V_{ϖ} \to R$ such that:

The map $t \to Φ (S (t) U)$ is nonincreasing for each $U \in V_{ϖ}$ .

If for some $U \in V_{ϖ}$ one has $Φ (S (t) U) = Φ (U)$ for all t, then $S (t) U = U$ for all $t ⩾ 0$ .

The above condition (ii) means that U must be a stationary point of $(V_{ϖ}, S (t))$ .

To prove the existence of a compact global attractor we rely on the following result (see [8] and references therein).

Theorem 4.1.

Assume that

The dynamical system $(V_{ϖ}, S (t))$ is asymptotically smooth.

The dynamical system $(V_{ϖ}, S (t))$ is gradient.

The Lyapunov functional $Φ (u)$ is bounded from above on any bounded subset of $V_{ϖ}$ .

The set $Φ_{R} = {U ∣ Φ (U) ⩽ R}$ is bounded for every R.

The set of stationary points $N$ is bounded.

Then the system $(V_{ϖ}, S (t))$ has a compact global attractor characterized by $A = M^{u} (N)$ .

An important characteristic of a global attractor is its fractal dimension.

Definition 4.3.

Let M be a compact set in a metric space X. The fractal dimension ${dim}_{f}^{X} M$ of M is defined by ${dim}_{f}^{X} M = \underset{ε \to 0}{lim sup} \frac{ln N (M, ε)}{ln (1 / ε)},$ where $N (M, ε)$ is the minimal number of closed sets of diameter ε which cover the set M.

Our proof of finite dimensionality of the attractors for $(V_{ϖ}, S (t))$ is based on the following assertion (see [7] and also [8] which contain other versions of the theorem stated below).

Theorem 4.2.

Let X be a Banach space and M be a bounded closed set in X. Assume that there exists a mapping $V : M \mapsto X$ such that $M \subseteq V M$ and

V is Lipschitz on M, i.e., there exists $L > 0$ such that $∥ V v_{1} - V v_{2} ∥ ⩽ L ∥ v_{1} - v_{2} ∥, v_{1}, v_{2} \in M;$

there exist compact seminorms $n_{1} (x)$ and $n_{2} (x)$ on X such that $∥ V v_{1} - V v_{2} ∥ ⩽ η ∥ v_{1} - v_{2} ∥ + K \cdot [n_{1} (v_{1} - v_{2}) + n_{2} (V v_{1} - V v_{2})]$ for any $v_{1}, v_{2} \in M$ , where $0 < η < 1$ and $K > 0$ are constants (a seminorm $n (x)$ on X is said to be compact iff $n (x_{m}) \to 0$ for any sequence ${x_{m}} \subset X$ such that $x_{m} \to 0$ weakly in X).

Then M is a compact set in X with finite fractal dimension. Moreover, if the seminorms

n_{1}

and

n_{2}

have the form

n_{i} (v) = ∥ ℘_{i} v ∥

i = 1, 2

, where

℘_{1}

and

℘_{2}

are finite-dimensional orthoprojectors, then,

{dim}_{f}^{X} M ⩽ (dim ℘_{1} + dim ℘_{2}) \cdot ln (1 + \frac{8 (1 + L) \sqrt{2} K}{1 - η}) \cdot {[ln \frac{2}{1 + η}]}^{- 1} .

To prove this we need to establish the following stabilizability estimate.

Theorem 4.3.

Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. Let $W (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν) = U^{1} (t) - U^{2} (t)$ be the difference of two weak solutions to problem ( 2.4 )–( 2.6 ) , $U^{1} (t) = (u^{1} (t), u_{t}^{1} (t), θ^{1} (t), η^{1}, P^{1} (t), ν^{1})$ and $U^{2} (t) = (u^{2} (t), u_{t}^{2} (t), θ^{2} (t), η^{2}, P^{2} (t), ν^{2})$ with initial data $U^{1} (0)$ and $U^{2} (0)$ respectively lying in a bounded positively invariant set $B$ . Then there exist constants $M > 0$ and $C_{B} > 0$ depending on $B$ such that ${∥ W (t) ∥}_{V_{ϖ}} ⩽ M e^{- c_{0} ϖ t} {∥ W (0) ∥}_{V_{ϖ}} + C_{B} max_{s \in [0, t]} ∥ A^{1 / 2} u (s) ∥$ (4.2) for all $t ⩾ 0$ , where $W (0) = U^{1} (0) - U^{2} (0)$ .

Proof.

It follows from the representation of the weak solutions to the nonlinear problem (2.26) that $W (t) = U (t) W (0) + \int_{0}^{t} U (t - s) (0, - f (u_{1} (s)) + f (u_{2} (s)), 0, 0, 0, 0) d s,$ where $U (t)$ is the evolution operator of the linear problem (2.4)–(2.6) (with $f \equiv 0$ ) (see Theorem 3.1). Observing Theorem 3.1, we have ${∥ W (t) ∥}_{V_{ϖ}} ⩽ M e^{- c_{0} ϖ t} {∥ W (0) ∥}_{V_{ϖ}} + M \int_{0}^{t} e^{- c_{0} ϖ (t - s)} ∥ - f (u^{1} (s)) + f (u^{2} (s)) ∥ d s .$

Since the initial data lie in a bounded positively invariant set $B$ we can conclude that there exists $C_{B} > 0$ depending on $B$ such that $∥ U^{1} (t) ∥_{V_{ϖ}} < C_{B}$ and $∥ U^{2} (t) ∥_{V_{ϖ}} < C_{B}$ for any $t ⩾ 0$ . Consequently, by (2.28) we get the following estimate ${∥ W (t) ∥}_{V_{ϖ}} ⩽ M e^{- c_{0} ϖ t} {∥ W (0) ∥}_{V_{ϖ}} + C_{B} \int_{0}^{t} e^{- c_{0} ϖ (t - s)} ∥ A^{1 / 2} u (s) ∥ d s .$ This yields the desired estimate (4.2). □

Theorem 4.4.

Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. The dynamical system $(V_{ϖ}, S (t))$ generated by problem ( 2.4 )–( 2.6 ) possesses a compact global attractor $A$ in $V_{ϖ}$ characterized by $A = M^{u} (N)$ .

Proof.

According to Theorem 4.1, we need to prove the assertions (i)–(v).

(i) The stabilizability inequality (4.2) can be written as follows ${∥ W (t) ∥}_{V_{ϖ}} ⩽ M e^{- c_{0} ϖ t} {∥ W (0) ∥}_{V_{ϖ}} + n^{T} (u)$ (4.3) for any $T > 0$ , where $n^{T} (u) = C_{B} {max}_{t \in [0, T]} ∥ u (t) ∥_{V_{1}}$ is a compact seminorm in $V_{2}$ since $V_{2} ↪ V_{1}$ compactly. Selecting T large enough for $M e^{- T}$ to be less than 1 we obtain the asymptotic smoothness of the dynamical system $(V_{ϖ}, S (t))$ by the version of the Ceron–Lopes theorem presented in [7] (see also [8]).

(ii) Let us take Φ as the energy functional E defined in (2.31). Then, for $U_{0} = (u_{0}, u_{1}, θ_{0}, η_{0}, P_{0}, ν_{0})$ , we infer from (2.30) and (2.10) that $\frac{d}{d t} Φ (S (t) U) = \int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s + \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s ⩽ 0,$ hence $Φ (S (t) z)$ is nonincreasing. Now let us suppose $Φ (S (t) U) = Φ (U)$ for all $t ⩾ 0$ . Then from the last inequality we discover that $\int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s + \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s = 0 \forall t ⩾ 0 .$ Both terms in the left-hand side of the above identity have the same sign and then $\int_{0}^{\infty} μ^{'} (s) {∥ B^{1 / 2} η (s) ∥}^{2} d s = \int_{0}^{\infty} λ^{'} (s) {∥ B^{1 / 2} ν (s) ∥}^{2} d s = 0 \forall t ⩾ 0 .$ Let $σ_{μ} = sup {s ∣ μ (s) > 0} and σ_{λ} = sup {s ∣ λ (s) > 0} .$ We note that $σ_{μ}$ (resp. $σ_{λ}$ ) may be equal to ∞. Then by force of $(H_{4})$ $μ^{'} (s) ⩽ - δ_{1} μ (s) < 0, s \in (0, σ_{μ}) and λ^{'} (s) ⩽ - δ_{2} λ (s) < 0, s \in (0, σ_{λ}),$ and therefore $\begin{array}{rcl} η (x, s) = 0 for a.e. (x, s) \in Ω \times (0, σ_{μ}), t ⩾ 0, \\ ν (x, s) = 0 for a.e. (x, s) \in Ω \times (0, σ_{λ}), t ⩾ 0 . \end{array}$ We plug $ϕ_{1} = ϕ_{2} = ϕ_{3} = ϕ_{4} = 0$ and any $ϕ_{5} \in C^{1} ([0, T]; M_{1, λ})$ such that $ϕ_{5} (T) = ϕ_{5} (0) = 0$ into the definition of a weak solution. We obtain $\int_{0}^{T} \int_{Ω} P ϕ_{5} d x d t = 0$ which implies that $P = 0$ . By using the same argument, we can obtain $θ = 0$ .

Now, if we plug $ϕ_{1} = ϕ_{2} = ϕ_{4} = ϕ_{5} = 0$ and any $ϕ_{3} \in C^{1} ([0, T]; V_{0})$ such that $ϕ_{3} (T) = ϕ_{3} (0) = 0$ into the definition of a weak solution, we obtain $\int_{0}^{T} \int_{Ω} Δ u ϕ_{3, t} d x d t = 0$ and, hence, for an arbitrary $ϕ_{3} \in H_{0}^{1} ([0, T]; H_{0}^{2} (Ω))$ , $\int_{0}^{T} \int_{Ω} u_{t} Δ ϕ_{3} d x d t = 0 .$ This implies that $u_{t} (t) \equiv 0$ and consequently $u (t) \equiv u_{0}$ for all $t ⩾ 0$ . So $S (t) U_{0} = U_{0} = (u_{0}, 0, 0, 0, 0, 0)$ .

(iii) Since the system $(V_{ϖ}, S (t))$ is gradient with Lyapunov functional $Φ = E$ , we deduce that $Φ (U)$ is bounded from above on bounded subsets $B$ of $V_{ϖ}$ .

(iv) Given $R > 0$ , the set $Φ_{R} = {U \in V_{ϖ} ∣ Φ (U) ⩽ R}$ is bounded. From (2.34) and (2.31), for all $U (t) \in Φ_{R}$ , we have ${∥ U (t) ∥}_{V_{ϖ}}^{2} ⩽ \frac{2}{℘} (Φ (U (t)) + c_{f} | Ω |) ⩽ 2 \frac{R + c_{f} | Ω |}{℘} .$

(v) Following the same argument as in [14], we consider the continuous functional $Π (u) : D (A) \to R$ defined by $Π (u) = \frac{m}{2} ∥ A u ∥_{2}^{2} + \int_{Ω} F (u) d x .$ Form (2.32), we infer $Π (u) ⩾ \frac{m}{2} ∥ A u ∥_{2}^{2} - c_{f} | Ω | .$ (4.4) Therefore, Π is bounded from below. Consider this functional on $℘_{k} D (A)$ , where $℘_{k}$ is the orthoprojector on $Span {e_{1}, \dots, e_{k}}$ , ${e_{i}}_{1}^{\infty}$ being an orthonormal basis of eigenvectors of A. Since $F (u) \to \infty$ , when $∥ A u ∥_{2}^{2} \to \infty$ , there exists a minimum point $u_{k}$ satisfying the equation $m A^{2} u_{k} = - ℘_{k} f (u_{k}) .$ (4.5) It thus follows from (4.4) that $\frac{m}{2} ∥ A u ∥_{2}^{2} ⩽ c_{f} | Ω | + inf_{u \in ℘_{k} D (A)} Π (u) ⩽ c_{f} | Ω | .$ (4.6) Then it follows from (2.14) and (4.5) that $∥ A^{2} u_{k} ∥ ⩽ C$ . Consequently, there exists a subsequence ${u_{k_{n}}}$ which converges weakly in $D (A^{2})$ to a function u. By the compactness result ${u_{k_{n}}}$ converges to u strongly in $D (A)$ . Then we can pass to the limit in (4.6) and it follows from (4.5) that u is a weak solution to the stationary problem. This implies that the set of stationary points is bounded.

We see that all the assertions of Theorem 4.1 are fulfilled and then the dynamical system $(V_{ϖ}, S (t))$ given by Theorem 2.1 has a compact global attractor $A = M^{u} (N)$ . □

Remark 4.1.

As mentioned above, every steady state has the form $U_{0} = (u_{0}, 0, 0, 0, 0, 0)$ , where $u_{0}$ solves the equation $m A^{2} u_{0} + f (u_{0}) = 0, x \in Ω,$ with boundary conditions $u_{0} = Δ u_{0} = 0, x \in Γ .$

Moreover, the global attractor $A$ consists of all full bounded trajectories $ϱ = {U (t) ∣ t \in R}$ such that [21] $lim_{t \to - \infty} {dist}_{X} (U (t), N) = 0 and lim_{t \to + \infty} {dist}_{X} (U (t), N) = 0 .$ A natural question is whether any point on the global attractor also has the form $U_{0} = (u_{0}, 0, 0, 0, 0, 0)$ (note that the set of steady states is a priori not finite, possibly not even discrete).

We now prove that the global attractor has finite fractal dimension and is a bounded subset of some “smoother” space.

Lemma 4.1.

Let us assume that conditions (1.4 ) , $(H_{1})$ – $(H_{5})$ and (2.13)–(2.15) hold. The compact global attractor $A$ in $V_{ϖ}$ given by Theorem 4.4 has finite fractal dimension.

Proof.

It follows from (4.3) that $max_{t \in [T, 2 T]} {∥ W (t) ∥}_{V_{ϖ}} ⩽ M e^{- c_{0} ϖ t} {∥ W (0) ∥}_{V_{ϖ}} + 2 T C_{B} max_{ξ \in [0, 2 T]} {∥ u (ξ) ∥}_{V_{1}} .$ (4.7)

Let $X = V_{2}$ , $Y = V_{1}$ , $Z = V_{0} \times M_{1, μ}$ and $Z^{'} = V_{0} \times M_{1, λ}$ be four reflexive Banach spaces with X compactly embedded into Y and put $V_{ϖ} = X \times Y \times Z \times Z^{'}$ , where the functions u, η and ν have the regularity $u \in C (R^{+}; X) \cap C^{1} (R^{+}; Y), η \in C (R^{+}; Z), ν \in C (R^{+}; Z^{'}) .$ Following [13] and [14] and references therein, we define the space $X = V_{ϖ} \times H (0, T)$ with an appropriate T, where $H (0, T) = {U (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν) \in C (0, T; V_{ϖ})}$ is equipped with the norm $\begin{matrix} {∥ U (t) ∥}_{H (0, T)} = & max_{[0, T]} (∥ u_{t} ∥^{2} + ϖ {∥ A^{1 / 2} u ∥}^{2} + m ∥ A u ∥^{2} + c ∥ θ ∥^{2} + 2 d ⟨ θ, P ⟩ \\ + r ∥ P ∥^{2} + ∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}) < \infty . \end{matrix}$ The norm in $X$ is given by ${∥ U (t) ∥}_{X} = {∥ U (0) ∥}_{V_{ϖ}} + {∥ U (t) ∥}_{H (0, T)},$ where $U = (U (0), U (t))$ . Now we consider the compact seminorm $n^{T} (W (t)) = C_{B} {max}_{ξ \in [0, T]} ∥ u (ξ) ∥_{V_{1}}$ as a seminorm on the space $X$ . Next we introduce the set $U_{T} = {U = (U (0), U (t)), t \in [0, T] ∣ U (0) \in A} .$ The operator $V_{T} : U_{T} \to X$ is defined by the formula $V_{T} (U (0), U (t)) = (U (T), U (t + T)) .$ We now verify that $V_{T}$ is Lipschitz continuous on $U_{T}$ . By the integral representation of the mild solution and Gronwall’s lemma we have that there exists $a > 0$ such that $\begin{matrix} {∥ V_{T} U^{1} - V_{T} U^{2} ∥}_{X} & = {∥ W (T) ∥}_{V_{ϖ}} + {∥ W (t + T) ∥}_{H (0, T)} \\ ⩽ e^{a t} ({∥ W (0) ∥}_{V_{ϖ}} + {∥ W (t) ∥}_{H (0, T)}) \\ ⩽ e^{a t} {∥ U^{1} - U^{2} ∥}_{X} \end{matrix}$ and we get the Lipschitz property for $V_{T}$ . From (4.7) we conclude that ${∥ V_{T} U^{1} - V_{T} U^{2} ∥}_{X} ⩽ M e^{- c_{0} ϖ T} {∥ U^{1} - U^{2} ∥}_{X} + C (n^{T} (U^{1} - U^{2}) + n^{T} (V_{T} U^{1} - V_{T} U^{2}))$ for all $U^{1}, U^{2} \in U_{T}$ , where $C > 0$ . At this point, we select T large enough to obtain $M e^{- c_{0} ϖ T} < 1$ . Thus, all conditions of Theorem 4.2 are satisfied and, therefore, ${dim}_{f}^{X} U_{T} < \infty$ . This implies that ${dim}_{f}^{V_{ϖ}} A < \infty$ . □

Remark 4.2.

Proving finite fractal dimension for our model is a very delicate matter, mainly due to the lack of compactness of the embedding $M_{p + 1, K} ↪ M_{p, K} .$

In what follows we prove that the compact global attractor $A$ in $V_{ϖ}$ given by Theorem 4.4 is a bounded subset of some “smoother” space. First, we note that by dissipativity the estimates $\begin{matrix} sup_{t \in R} {∥ u_{t} ∥^{2}, ϖ {∥ A^{1 / 2} u ∥}^{2}, ∥ A u ∥^{2}, ∥ θ ∥^{2}, ∥ P ∥^{2}} ⩽ C_{R}, \\ sup_{t \in R} {∥ η ∥_{M_{1, μ}}^{2} + ∥ ν ∥_{M_{1, λ}}^{2}} ⩽ C_{R} \end{matrix}$ (4.8) hold true for any trajectory ${U (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν)} \subset V_{ϖ}$ lying in the attractor $A$ .

Let us introduce the space $W_{ϖ} = {u \in D (A) ∣ A^{2} u \in V_{ϖ}^{'}},$ (4.9) where $V_{ϖ}^{'}$ is dual to $V_{ϖ} = D (M_{ϖ}^{1 / 2})$ . It is readily verified that $W_{ϖ} = D (A^{3 / 2}) = {u \in H^{3} (Ω) ∣ u = Δ u = 0 on Γ} .$ (4.10)

By using similar arguments as in [7–10] and references therein, we prove in the following result that the attractor is a bounded subset of some “smoother” space.

Theorem 4.5.

Let us assume that conditions ( 1.4 ) , $(H_{1})$ – $(H_{5})$ and ( 2.13 )–( 2.15 ) hold. The attractor $A$ is a bounded subset of the space $Y = W_{ϖ} \times D (A) \times D (A) \times M_{1, μ} \times D (A) \times M_{1, λ} .$ Moreover, there exists a positive constant C not depending on t and ϖ such that for any trajectory ${U (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν)} \subset V_{ϖ}$ lying in the attractor $A$ we have $\begin{array}{rclr} sup_{t \in R} {∥ u ∥_{3}^{2} + ∥ u_{t t} ∥^{2} + ϖ {∥ A^{1 / 2} u_{t} ∥}^{2} + ∥ A u_{t} ∥^{2} + ∥ θ_{t} ∥^{2} + ∥ P_{t} ∥^{2} + ∥ η_{t} ∥_{M_{1, μ}}^{2} + ∥ ν_{t} ∥_{M_{1, λ}}^{2}} \\ ⩽ C_{R}, & (4.11) \\ sup_{t \in R} {∥ L_{μ} η ∥^{2}} ⩽ C_{R}, sup_{t \in R} {∥ L_{λ} ν ∥^{2}} ⩽ C_{R} & (4.12) \end{array}$ and $∥ θ ∥_{D (A)} ⩽ C_{R}, ∥ P ∥_{D (A)} ⩽ C_{R},$ (4.13) where the estimates are uniform in $0 < ϖ ⩽ 1$ .

Proof.

Let ${U (t) = (u (t), u_{t} (t), θ (t), η, P (t), ν)} \subset V_{ϖ}$ be a full trajectory from the attractor $A$ . Applying (4.3) with $U^{1} = U (s + ω)$ , $U^{2} = U (s)$ (and, accordingly, the interval $[s, t]$ in place of $[0, t]$ ), we have $\begin{array}{rclr} {∥ U (t + ω) - U (t) ∥}_{V_{ϖ}}^{2} ⩽ M e^{c_{0} ϖ (t - s)} {∥ U (s + ω) - U (s) ∥}^{2} + C_{B} max_{ξ \in [s, t]} {∥ u (ξ + ω) - u (ξ) ∥}_{V_{1}}^{2} & (4.14) \end{array}$ for any $t, s \in R$ such that $s ⩽ t$ and for any ω with $| ω | < 1$ . Taking the limit $s \to - \infty$ , (4.14) gives ${∥ U (t + ω) - U (t) ∥}_{V_{ϖ}}^{2} ⩽ C_{B} max_{ξ \in [- \infty, t]} {∥ u (ξ + ω) - u (ξ) ∥}^{2}$ (4.15) for any $t \in R$ and $| ω | < 1$ . On the other hand, on the attractor we have that $\frac{1}{ω} ∥ u (ξ + ω) - u (ξ) ∥ ⩽ \frac{1}{ω} \int_{0}^{ω} ∥ u_{t} (ξ + t) ∥ d t, t \in R .$ Therefore, by (2.35) we obtain $max_{ξ \in R} {∥ \frac{U (ξ + ω) - U (ξ)}{ω} ∥}_{V_{ϖ}} ⩽ C_{R}, | ω | < 1 .$ The last estimate implies that the function $U (t)$ is absolutely continuous and thus possesses a derivative almost everywhere which satisfies ${∥ U_{t} (t) ∥}_{V_{ϖ}} ⩽ C_{R} .$ (4.16)

The above estimate together with (2.4)_2,3 enables us to establish $sup_{t \in R} {∥ L_{μ} η ∥^{2}} ⩽ C (∥ θ_{t} ∥^{2} + ∥ P_{t} ∥^{2} + ∥ A u_{t} ∥^{2}) ⩽ C_{R}$ (4.17) and $sup_{t \in R} {∥ L_{λ} ν ∥^{2}} ⩽ C (∥ θ_{t} ∥^{2} + ∥ P_{t} ∥^{2} + ∥ A u_{t} ∥^{2}) ⩽ C_{R} .$ (4.18)

In view of (2.4)_2–5 and (4.16) it is easily seen that on the attractor one has $\begin{matrix} \int_{0}^{\infty} μ (s) B η (s) d s = - c θ_{t} - d P_{t} + γ_{1} Δ u_{t} = Ψ_{1}, \\ \int_{0}^{\infty} λ (s) B ν (s) d s = - d θ_{t} - r P_{t} + γ_{2} Δ u_{t} = Ψ_{2}, \end{matrix}$ (4.19) where $∥ Ψ_{1} ∥ ⩽ C_{R}, ∥ Ψ_{2} ∥ ⩽ C_{R} .$ (4.20) Substituting (2.23) into (4.19) we get $\begin{matrix} c_{μ} B θ (t) = Ψ_{1} - \int_{0}^{\infty} μ (s) \int_{0}^{s} e^{τ - s} B η^{*} (τ) d τ d s, \\ c_{λ} B P (t) = Ψ_{2} - \int_{0}^{\infty} λ (s) \int_{0}^{s} e^{τ - s} B ν^{*} (τ) d τ d s, \end{matrix}$ (4.21) where $η^{*}$ , $ν^{*}$ , $c_{μ}$ and $c_{λ}$ are defined by (2.20)₄, (2.20)₆, (2.23)₁ and (2.23)₄, respectively. In view of (2.24) and (4.20), it is easily seen that $∥ B θ (t) ∥ ⩽ C_{R}, ∥ B P (t) ∥ ⩽ C_{R} .$ (4.22) Since $B = h I + A$ , we infer from (4.22) and (4.8) that $∥ A θ (t) ∥ ⩽ C_{R}, ∥ A P (t) ∥ ⩽ C_{R} .$ (4.23)

Finally, since $∥ A^{- 1 / 2} M_{ϖ}^{1 / 2} ∥ ⩽ C$ we can deduce from equation (2.4)₁ that $∥ A^{- 1 / 2} Δ^{2} u ∥ ⩽ ∥ M_{ϖ}^{- 1 / 2} Δ^{2} u ∥ ⩽ C (∥ M_{ϖ}^{1 / 2} u_{t t} ∥ + ∥ A θ ∥ + ∥ A P ∥ + ∥ F (u) ∥) ⩽ C .$ (4.24) Thus, we can conclude that $t \mapsto u (t)$ is a bounded function in the space $W_{ϖ}$ defined in (4.9).

The final conclusion needed for (4.11) follows by realizing that $∥ u ∥_{3} ⩽ C ∥ M_{ϖ}^{- 1 / 2} A_{1} u ∥ uniformly in 0 < ϖ ⩽ 1,$ where $A_{1}$ is the biharmonic operator defined in (2.8). Indeed, this last assertion follows from the facts that $M_{ϖ}^{- 1 / 2} A_{1} u$ is bounded on $L^{2} (Ω)$ with a norm independent of $ϖ > 0$ (see [9]) and $D (A_{1}^{3 / 4}) \subset H^{3} (Ω)$ (see [20]).

Estimates (4.16) and (4.24) entail (4.11). Estimate (4.12) immediately follows from (4.17) and (4.18). Finally, estimate (4.13) is a straightforward corollary of (4.23). The proof is complete. □

Remark 4.3.

By following the approach developed in [12], one can obtain more regularity on the memory components by proving that the variables η and ν are bounded in more regular spaces. This requires further (and rather long) calculations which are omitted here for the sake of brevity. We refer the interested reader to [12] for details.

The existence of the global attractor is proved in [4] for $ϖ = 0$ and we should also add, in that case, $∥ u ∥_{4}^{2} ⩽ C_{R}$ in Theorem 4.5.

Footnotes

Acknowledgements

The authors would like to thank the referees for their critical review and valuable comments which allowed to improve the paper. The first author would like to thank the University of Poitiers and the “Laboratoire de Mathématiques et Applications” for their hospitality and support during his stay at Poitiers in the frame of the Erasmus Mundus-Al Idrisi II program. The work of M. Aouadi was partially supported by the Erasmus Mundus-Al Idrisi II program.

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Smooth attractor for a nonlinear thermoelastic diffusion thin plate based on Gurtin–Pipkin’s model

Abstract

Keywords

1. Introduction

Footnotes

Acknowledgements

References