Study of the dissipativity,global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Abstract

In this paper, we study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms, with initial and homogenous Dirichlet boundary condition.

Keywords

Caginalp hyperbolic phase field singular potential Dirichlet boundary bounded absorbing set dissipativity global attractor exponential attractor

1. Introduction

The order parameter theory applied by L. D. Landau to critical phenomena made it possible to calculate the exponents for which the thermodynamic quantities diverge at the critical point (for example, the temperature at which the distinction between the liquid state and the gaseous state disappears). This corresponds to a truncation of the higher modes of Fourier. In particular, the order parameter theory leads, in critical phenomena, to calculations of exponents with the right order of magnitude, but does not allow to have the correct numerical values (see [8]).

The question is whether the order parameter in critical phenomena can be used to solve dynamic problems far from the critical point, as is the case for an ordinary phase transition (this is particularly the case for a solid-liquid transition). A satisfactory answer to this question was given by G. Caginalp who showed that the reason for neglecting the higher Fourier modes comes from the fact that the transition length in the order parameter is very small compared to the relevant macroscopic scales (see [3,8]).

In this article, we are interested in the study of the following hyperbolic relaxation of the Caginalp phase-field system which is based on a variant of the Maxwell–Cattaneo law for thermal conduction rather than Fourier’s law, in a bounded smooth domain Ω ⊂ $R^{n} (1 ⩽ n ⩽ 3)$ . $\begin{array}{l} (1) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u + f (u) = α, \\ (2) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{l} (3) & u |_{\partial Ω} = α |_{\partial Ω} = 0, \end{array}$ and initial conditions $\begin{array}{l} (4) & u |_{t = 0} = u (0), \frac{\partial u}{\partial t} |_{t = 0} = u_{1}, α |_{t = 0} = α (0), \frac{\partial α}{\partial t} |_{t = 0} = α_{1}, \end{array}$ where $ϵ > 0$ , is a relaxation parameter, $u = u (t, x)$ the order parameter, $α = α (t, x)$ the relative temperature, with $α (t, x) = \int_{0}^{t} θ (x, τ) d τ + α (0)$ , f is a given singular potential function.

f satisfies the following assumptions: $\begin{array}{l} (5) & f \in C^{3} (] - 1, 1 [), lim_{s \to \pm 1} f (s) = \pm \infty, lim_{s \to \pm 1} f^{'} (s) = + \infty, f (0) = 0, \\ (6) & - c_{0} ⩽ F (s) ⩽ f (s) s + c_{0}, c_{0} ⩾ 0, for all s \in] - 1, 1 [, where F (s) = \int_{0}^{s} f (τ) d τ, \end{array}$ and $\begin{array}{l} (7) & f^{'} (s) ⩾ - c_{1}, c_{1} ⩾ 0, for all s \in] - 1, 1 [. \end{array}$ We recall that a typical example of function f is: $\begin{array}{l} f (s) = k_{1} ln (\frac{1 + s}{1 - s}) - k_{0} s, for all s \in] - 1, 1 [, with 0 < k_{1} < k_{0} . \end{array}$

Such studies had been carried already out in several works with various types of the conditions at the boundary and regular and not regular potential functions; to see for example [4–8,11–14].

Very recently, this study was realised in the same case of a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms with homogenous Dirichlet boundary conditions, in a bounded smooth domain and the same case initial conditions with the aim of studying the existence and the uniqueness of this system (see [9]).

When $ϵ = 0$ , the system is parabolic-hyperbolic, the existence of the solution as well as that of the global attractor are proved by classical methods (see [8]).

Our main objective in this article is to study dissipativity, to prove the existence of the global attractor and to show that it is of finite fractal dimension, by establishing the existence of an exponential attractor, which gives a upper bound of the dimension of the global attractor. For this, we show that the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , associated with our system has the regularization property of the semigroup, is Hölder continuous in time (see [1,2] and [15]) and that it also verifies a continuous Lipschitz. To overcome the difficulty preventing us from using classical methods, since when $ϵ > 0$ , the system is hyperbolic, we cannot use classical methods, we will then decompose the difference of two trajectories of the problem into two parts, the first part tends towards zero when time tends towards infinity, and the second is a compacting part for positive t (see [10]) as follows: in Section 1, we are talking about the introduction, in Section 2, we present the notations used in the article, in Section 3, we present the main results (see [9]), in Section 4, we study the dissipativity in term of absorbing bounded sets for the semigroup, in Section 5, we study the global attractor. Finally, in Section 6, we study the exponential attractor as follows: in subsection 6.1, we have the regularization property of the semigroup, in subsection 6.2, then we have the Hölder continuous in time and finally, in subsection 6.3, we study the continuous Lipschitz.

2. Notations

In this section, we define some mathematical quantities which are useful in our work.

We introduce the following quantity: $\begin{array}{l} D [u (t)] = {(1 - {‖ u (t) ‖}_{L^{\infty}})}^{- 1} . \end{array}$ We also introduce the standard energy norm $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{k} (ϵ)}^{2} = {‖ u (t) ‖}_{H^{k + 1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{k}}^{2} + {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{k - 1}}^{2} . \end{array}$ Thus, the space $ε^{k} (ϵ)$ coincides with $[H^{k + 1} (Ω) \times H^{k} (Ω)] \cap {ζ |_{\partial Ω} = 0}$ if $ϵ > 0$ and with $[H^{k + 1} (Ω) \times H^{k - 1} (Ω)] \cap {ζ |_{\partial Ω} = 0}$ if $ϵ = 0$ , whenever the traces make sense.

We denote by $‖ \cdot ‖$ and $(\cdot, \cdot)$ (or $‖ \cdot ‖_{Φ}$ and ${(\cdot, \cdot)}_{Φ}$ ) the norm and the scalar product in $L^{2} (Ω)$ (in Φ).

We introduce Q the monotonic function are independent of ϵ.

We have thanks to the Theorem 3.1 (see [9]) and the Theorem 3.2 (see [9]), we have following phase spaces: $Φ_{k} = {(ζ_{u} (0), α, v) \in ε^{k} (ϵ) \times (H^{k + 1} (Ω) \cap H_{0}^{1} (Ω)) \times (H^{k} (Ω) \cap H_{0}^{1} (Ω)) : ‖ u_{0} ‖_{L^{\infty} (Ω)} < 1}$ , for $k = 1, 2$ .

Moreover, we define the continuous semigroup $S_{ϵ} (t)$ for all $t ⩾ 0$ , as follows: $\begin{array}{l} S_{ϵ} (t) : Φ_{k} ⟶ Φ_{k}, for k = 1, 2, \\ (ζ_{u} (0), α_{0}, α_{1}) ⟼ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}), \end{array}$ with $(ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t})$ such that $(u, α)$ is the unique solution of problem (1)–(4) with initial conditions $(ζ_{u} (0), α_{0}, α_{1}) \in Φ_{k}$ (see [9]).

3. Main results

We have showed the existence, the uniqueness solution and the regularity of the solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms (see [9]) as follows:

Theorem 3.1 (See [9]).

Let $ϵ ⩽ ϵ_{0}$ and f nonlinearity satisfying ( 5 ). Then there is a decreasing function $R : (0, ϵ_{0}] ⟶ R_{+}$ such that $\begin{array}{l} (8) & lim_{ϵ \to 0} R (ϵ) = + \infty \end{array}$ and for every initial data $(ζ_{u} (0), α (0), α_{1})$ satisfying $\begin{array}{l} (9) & D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2} ⩽ R (ϵ), \end{array}$ the system ( 1 )–( 2 ) has a unique global solution $(u, α)$ such that $u, α \in L^{\infty} (0, T; H^{2} (Ω) \cap H_{0}^{1} (Ω))$ sufficiently regular which is separated from the singular points, i.e. $\begin{array}{l} (10) & ‖ u ‖_{L^{\infty} ((0, T) \times Ω)} < 1, \end{array}$ satisfies the following estimate $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ \\ (11) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

Theorem 3.2 (See [9]).

Let $ϵ ⩽ 1$ and $(u, α)$ a solution of problem ( 1 )–( 4 ) such that $(ζ_{u} (0), α (0), α_{1}) \in Φ_{2}$ . Then, the solution $(u, α)$ satisfies the following estimate: $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (12) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, \end{array}$

where C and β positive constants and the monotonic function Q are independent of ϵ.

4. Dissipativity

We study here, the existence of bounded sets absorbing for the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , associated with our problem (1)–(4). Thus, we have the following propositions:

Proposition 4.1.
The semigroup of operators $S_{ϵ} (t)$ associated with the problem ( 1 )–( 4 ) is dissipative in $Φ_{1}$ ; i.e, it possesses a bounded set absorbing in $Φ_{1}$ .
Proof.
We consider $B_{1} = {(ζ_{u}, α, \frac{\partial α}{\partial t}) \in Φ_{1} / D [u] + ‖ (ζ_{u}, α, \frac{\partial α}{\partial t}) ‖_{Φ_{1}} ⩽ R_{1}}$ a bounded set in $Φ_{1}$ and $R_{1} > 0$ .

We set $E (R_{1}) = Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{1}}^{2})$ , independent of ϵ. We have $t_{1} = t_{1} (R_{1}) = max (0, - \frac{1}{β} ln (\frac{C}{E (R_{1}}))$ , $β > 0$ , where C positive constant and the monotonic function Q are independent of ϵ.

Using the estimate (11) (see [9]) and for all $t ⩾ t_{1}$ , we have $\begin{array}{l} {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{1}}^{2} ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t} + C, \end{array}$ then $\begin{array}{l} {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{1}} & ⩽ \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t_{1}} + C} \\ ⩽ \sqrt{E (R_{1}) + C}, \end{array}$ and $\begin{array}{l} D [u (t)] & ⩽ (Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t} + C) \\ ⩽ (Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t_{1}} + C) \\ ⩽ E (R_{1}) + C, \end{array}$ therefore $\begin{array}{l} D [u (t)] + {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{1}} \\ ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) + C + \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) + C}, \end{array}$ which implies $\begin{array}{l} D [u (t)] + {‖ S_{ϵ} (t) (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}} \\ ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) + C \\ (13) & + \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2}) + C} . \end{array}$ Finally, the semigroup $S_{ϵ} (t)$ admets a bounded set absorbing the $B_{R_{1}} (ϵ)$ bowl of O center and $R_{1} = Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{1}}^{2}) + C + \sqrt{Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{1}}^{2}) + C}$ ray.

Where C positive constant and the monotonic function Q are independent of ϵ. □
Proposition 4.2.
The semigroup of operators $S_{ϵ} (t)$ associated with the problem ( 1 )–( 4 ) is dissipative in $Φ_{2}$ ; it possesses a bounded set absorbing in $Φ_{2}$ .
Proof.
We consider $B_{2} = {(ζ_{u}, α, \frac{\partial α}{\partial t}) \in Φ_{2} / D [u] + ‖ (ζ_{u}, α, \frac{\partial α}{\partial t}) ‖_{Φ_{2}} ⩽ R_{2}}$ a bounded set in $Φ_{2}$ and $R_{2} > 0$ .

We set $E (R_{2}) = Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{2}}^{2})$ , independent of ϵ. We have $t_{2} = t_{2} (R_{2}) = max (0, - \frac{1}{β} ln (\frac{C}{E (R_{2}}))$ , $β > 0$ , where C positive constant and the monotonic function Q are independent of ϵ.

Using (12) (see [9]) and for all $t ⩾ t_{2}$ , we have $\begin{array}{l} {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{2}}^{2} ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) e^{- β t} + C, \end{array}$ then $\begin{array}{l} {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{2}} & ⩽ \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) e^{- β t_{1}} + C} \\ ⩽ \sqrt{E (R_{2}) + C}, \end{array}$ and $\begin{array}{l} D [u (t)] & ⩽ (Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) e^{- β t} + C) \\ ⩽ (Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) e^{- β t_{1}} + C) \\ ⩽ E (R_{2}) + C, \end{array}$ therefore $\begin{array}{l} D [u (t)] + {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{2}} \\ ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) + C + \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) + C}, \end{array}$ which implies $\begin{array}{l} D [u (t)] + {‖ S_{ϵ} (t) (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}} \\ ⩽ Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) + C \\ (14) & + \sqrt{Q (D [u (0)] + {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{2}}^{2}) + C} . \end{array}$ Finally, the semigroup $S_{ϵ} (t)$ admets a bounded set absorbing the $B_{R_{2}} (ϵ)$ bowl of O center and $R_{2} = Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{2}}^{2}) + C + \sqrt{Q (D [u (0)] + ‖ (ζ_{u} (0), α (0), α_{1}) ‖_{Φ_{2}}^{2}) + C}$ ray.

Where C positive constant and the monotonic function Q are independent of ϵ. □

5. The global attractor

The global attractor is the smallest set (for the inclusion) compact of the phases space which is invariant by the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , and attracts all bounded sets of the initial data when time t goes to the infinity.

Theorem 5.1.
Assume the hypotheses of Theorem 3.1 (see [ 9 ]) and of the Proposition 4.1 , the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , defined from $Φ_{1}$ in $Φ_{1}$ , and associated on the problem ( 1 )–( 4 ) possesses the global attractor in $Φ_{1}$ , bounded and connexe in $Φ_{2}$ .
Proof.
Let us break up the semigroup as the sum of two operators family $S_{ϵ}^{1} (t)$ and $S_{ϵ}^{2} (t)$ , $t ⩾ 0$ the first family tending towards zero when t tends towards the infinity and the second family is regularizing.

In other words, $\begin{array}{l} S_{ϵ} (t) = S_{ϵ}^{1} (t) + S_{ϵ}^{2} (t), \end{array}$ where the operator $S_{ϵ}^{1} (t)$ is defined by: $\begin{array}{l} S_{ϵ}^{1} (t) : Φ_{1} ⟶ Φ_{1}, \\ (ζ_{v} (0), η (0), \frac{\partial η (0)}{\partial t}) ⟼ (ζ_{v} (t), η (t), \frac{\partial η (t)}{\partial t}), \end{array}$ tends towards 0 lorsque t tends towards the infinity, with the operator $S_{ϵ}^{2} (t)$ is defined by: $\begin{array}{l} S_{ϵ}^{2} (t) : Φ_{2} ⟶ Φ_{2}, \\ (ζ_{w} (0), ξ (0), \frac{\partial ξ (0)}{\partial t}) ⟼ (ζ_{w} (t), ξ (t), \frac{\partial ξ (t)}{\partial t}), \end{array}$ is regularizing.

We consider the following decomposition: $\begin{array}{l} (u, α) = (v, η) + (w, ξ), \end{array}$ where $(v, η)$ is solution of the linearity system $\begin{array}{l} (15) & ϵ \frac{\partial^{2} v}{\partial t^{2}} + \frac{\partial v}{\partial t} - Δ v = η, \\ (16) & \frac{\partial^{2} η}{\partial t^{2}} + \frac{\partial η}{\partial t} - Δ \frac{\partial η}{\partial t} - Δ η = - v - \frac{\partial v}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{l} (17) & ζ_{v} |_{\partial Ω} = η |_{\partial Ω} = 0, \end{array}$ and initial conditions $\begin{array}{l} (18) & ζ_{v} |_{t = 0} = ζ_{v} (0), η |_{t = 0} = α_{0}, \frac{\partial η}{\partial t} |_{t = 0} = α_{1}, \end{array}$ with $(w, ξ)$ is solution of the nonlinearity system $\begin{array}{l} (19) & ϵ \frac{\partial^{2} w}{\partial t^{2}} + \frac{\partial w}{\partial t} - Δ w + f (u) = ξ, \\ (20) & \frac{\partial^{2} ξ}{\partial t^{2}} + \frac{\partial ξ}{\partial t} - Δ \frac{\partial ξ}{\partial t} - Δ ξ = - w - \frac{\partial w}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{l} (21) & ζ_{w} |_{\partial Ω} = ξ |_{\partial Ω} = 0, \end{array}$ and initial conditions $\begin{array}{l} (22) & ζ_{w} |_{t = 0} = ξ |_{t = 0} = \frac{\partial ξ}{\partial t} |_{t = 0} = 0 . \end{array}$ We now do a certain number of estimates a priori.

Multipling (15) by $- Δ \frac{\partial v}{\partial t}$ and integrating over Ω, we have $\begin{array}{l} (23) & \frac{d}{d t} (ϵ {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + ‖ Δ v ‖^{2}) + {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} ⩽ 2 ‖ \nabla η ‖^{2} . \end{array}$ Multipling (15) by $- Δ v$ and integrating over Ω, we have $\begin{array}{l} (24) & \frac{d}{d t} (‖ \nabla v ‖^{2} + 2 ϵ (\nabla \frac{\partial v}{\partial t}, \nabla v)) + ‖ Δ v ‖^{2} ⩽ 2 {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + 2 ‖ η ‖^{2} . \end{array}$ Multipling (16) by $- Δ η$ and integrating over Ω, we have $\begin{array}{l} (25) & \frac{d}{d t} (‖ \nabla η ‖^{2} + ‖ Δ η ‖^{2} + 2 (\nabla \frac{\partial η}{\partial t}, \nabla η)) + ‖ Δ η ‖^{2} ⩽ 2 {‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + 2 ‖ v ‖^{2} + 2 {‖ \frac{\partial v}{\partial t} ‖}^{2} . \end{array}$ Multipling (16) by $- Δ \frac{\partial η}{\partial t}$ and integrating over Ω, we have $\begin{array}{l} (26) & \frac{d}{d t} ({‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + ‖ Δ η ‖^{2}) + {‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial η}{\partial t} ‖}^{2} ⩽ 2 ‖ v ‖^{2} + 2 {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{8}$ (23), $γ_{9}$ (24), $γ_{10}$ (25) and $γ_{11}$ (26) with $γ_{8} > 0$ , $γ_{9} > 0$ , $γ_{10} > 0$ , $γ_{11} > 0$ such that $\begin{array}{l} γ_{8} - 2 γ_{9} - 2 c_{p} γ_{10} - 2 γ_{11} > 0, \\ γ_{9} - 2 c_{p} C_{3} γ_{10} - 2 c_{p} C_{4} γ_{11} > 0, \\ γ_{10} - 2 C_{1} γ_{8} - c_{p} C_{2} γ_{9} > 0, \end{array}$ we have $\begin{array}{l} (27) & \frac{d}{d t} E_{3} + C_{1} ‖ Δ η ‖^{2} + C_{2} {‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + C_{3} ‖ Δ v ‖^{2} + C_{4} {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + C_{5} {‖ Δ \frac{\partial η}{\partial t} ‖}^{2} ⩽ 0, \end{array}$ where $\begin{array}{l} E_{3} = & γ_{8} (ϵ {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + ‖ Δ v ‖^{2}) + γ_{9} (‖ \nabla v ‖^{2} + 2 ϵ (\nabla \frac{\partial v}{\partial t}, \nabla v)) \\ + γ_{10} (‖ \nabla η ‖^{2} + ‖ Δ η ‖^{2} + 2 (\nabla \frac{\partial η}{\partial t}, \nabla η)) + γ_{11} ({‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + ‖ Δ η ‖^{2}) . \end{array}$ We have $\begin{array}{l} (28) & γ_{8} ϵ {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + γ_{9} (‖ \nabla v ‖^{2} + 2 ϵ (\nabla \frac{\partial v}{\partial t}, \nabla v)) ⩾ ϵ (γ_{8} - 2 γ_{9}) {‖ \nabla \frac{\partial v}{\partial t} ‖}^{2} + \frac{γ_{9}}{2} ‖ \nabla v ‖^{2}, \end{array}$ and $\begin{array}{l} (29) & γ_{11} {‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + γ_{10} (‖ \nabla η ‖^{2} + 2 (\nabla \frac{\partial η}{\partial t}, \nabla η)) ⩾ (γ_{11} - 2 γ_{10}) {‖ \nabla \frac{\partial η}{\partial t} ‖}^{2} + \frac{γ_{10}}{2} ‖ \nabla η ‖^{2} . \end{array}$ Choosing $γ_{8}$ , $γ_{9}$ , $γ_{10}$ and $γ_{11}$ such that $\begin{array}{l} γ_{8} - 2 γ_{9} > 0, \\ γ_{11} - 2 γ_{10} > 0, \end{array}$ there exist a positive constant C independent of ϵ such that $\begin{array}{l} C^{- 1} ({‖ v (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η (t)}{\partial t} ‖}_{H^{1}}^{2}) ⩽ E_{3} (t) \\ (30) & ⩽ C ({‖ v (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η (t)}{\partial t} ‖}_{H^{1}}^{2}) . \end{array}$ We end up using the relation (30) to an estimate of the following form $\begin{array}{l} (31) & \frac{d}{d t} E_{3} + β E_{3} + C_{1} {‖ \frac{\partial v}{\partial t} ‖}_{H^{1}}^{2} + C_{2} {‖ \frac{\partial η}{\partial t} ‖}_{H^{1}}^{2} ⩽ 0, C_{i} > 0, \end{array}$ where β and C are positive constants independent of ϵ.

Applying Gronwall’s lemma at the estimate (31), we obtain $\begin{array}{l} (32) & E_{3} (t) + \int_{0}^{t} ({‖ \frac{\partial v (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial η (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ ⩽ E_{3} (0) e^{- β t} . \end{array}$ Then, using the estimate (30) we deduce $\begin{array}{l} {‖ v (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (33) & + \int_{0}^{t} ({‖ \frac{\partial v (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial η (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t}, \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

Let us consider the following equation $\begin{array}{l} (34) & ϵ \frac{\partial^{2} v}{\partial t^{2}} + \frac{\partial v}{\partial t} - Δ v = h (t) = η, \\ (35) & ζ_{v} (t) |_{\partial Ω} = h (t) |_{\partial Ω} = 0 . \end{array}$ The application of the Proposition A.2 (see [6]) to equation (34) gives $\begin{array}{l} {‖ ζ_{v} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial v (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ C ({‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ h (0) ‖}^{2}) e^{- β t} \\ (36) & + C \int_{0}^{t} ({‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2}) e^{- β (t - τ)} d τ . \end{array}$ Let us determine an estimate of $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ We have $\begin{array}{l} (37) & {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} = {‖ η (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial η (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ Let us determine the estimates $‖ η (τ) ‖_{H^{1}}$ and $‖ \frac{\partial η (τ)}{\partial t} ‖_{H^{- 1}}^{2}$ . $\begin{array}{l} (38) & {‖ η (τ) ‖}_{H^{1}} ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{\frac{- β τ}{2}}, \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

And $\begin{array}{l} (39) & {‖ \frac{\partial η (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{\frac{- β τ}{2}}, \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

The estimate becomes $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β τ}, \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

Thanks to the estimate above, the estimate (36) becomes $\begin{array}{l} {‖ ζ_{v} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial v (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t} + \int_{0}^{t} Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β τ} d τ \\ (40) & ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t}, for all t \in [0, T], \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

Therefore $\begin{array}{l} {‖ S_{ϵ}^{1} (t) (ζ_{v} (0), η (0), \frac{\partial η (0)}{\partial t}) ‖}_{Φ_{1}}^{2} + \int_{0}^{t} ({‖ \frac{\partial v (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial η (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ \\ (41) & ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t}, \end{array}$ where β positive constant and the monotonic function Q are independent of ϵ.

We see that the operators $S_{ϵ}^{1} (t)$ tend to zero when t tends to infinity (uniformly compared to bounded sets of initial conditions).

We now consider the problem (19)–(22).

Multipling (19) by $Δ^{2} \frac{\partial w}{\partial t}$ and integrating over Ω, we have $\begin{array}{l} (42) & \frac{d}{d t} (ϵ {‖ \frac{\partial w}{\partial t} ‖}_{H^{2}}^{2} + ‖ w ‖_{H^{3}}^{2}) + \frac{1}{2} {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} ⩽ C_{1} ‖ Δ u ‖^{2} + C_{2} ‖ \nabla Δ ξ ‖^{2}, C_{i} > 0 . \end{array}$

Multipling (19) by $- Δ \frac{\partial^{2} w}{\partial t^{2}}$ and integrating over Ω, we have $\begin{array}{l} (43) & \frac{d}{d t} {‖ \frac{\partial w}{\partial t} ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial^{2} w}{\partial t^{2}} ‖}_{H^{1}}^{2} ⩽ C_{1} ‖ \nabla Δ w ‖^{2} + C_{2} ‖ Δ v ‖^{2} + C_{3} ‖ Δ \nabla ξ ‖^{2} . \end{array}$

Multipling (19) by $Δ^{2} w$ and integrating over Ω, we have $\begin{array}{l} (44) & \frac{d}{d t} (‖ Δ w ‖^{2} + 2 ϵ (Δ \frac{\partial w}{\partial t}, Δ w)) + ‖ \nabla Δ w ‖^{2} ⩽ {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} + C_{4} ‖ Δ u ‖^{2} + C_{5} ‖ \nabla Δ ξ ‖^{2} . \end{array}$ Multipling (20) by $Δ^{2} \frac{\partial ξ}{\partial t}$ and integrating over Ω, we have $\begin{array}{l} (45) & \frac{d}{d t} (‖ ξ ‖_{H^{3}}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{2}}^{2}) + 2 {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{2}}^{2} ⩽ C ‖ \nabla Δ w ‖^{2} + 2 {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} . \end{array}$ Multipling (20) by $Δ^{2} ξ$ and integrating over Ω, we have $\begin{array}{l} \frac{d}{d t} (‖ Δ ξ ‖^{2} + ‖ \nabla Δ ξ ‖^{2} + 2 (Δ \frac{\partial ξ}{\partial t}, Δ ξ)) + ‖ \nabla Δ ξ ‖^{2} \\ (46) & ⩽ 2 {‖ Δ \frac{\partial ξ}{\partial t} ‖}^{2} + C_{6} ‖ \nabla Δ w ‖^{2} + C_{7} {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{12}$ (42), $γ_{13}$ (43), $γ_{14}$ (44), $γ_{15}$ (45) and $γ_{16}$ (46), with $γ_{i} > 0$ such that $\begin{array}{l} \frac{γ_{12}}{2} - γ_{14} - 2 γ_{15} - C_{7} γ_{16} > 0, \\ γ_{16} - C_{5} γ_{14} - C_{3} γ_{13} - C_{2} γ_{12} > 0, \\ γ_{14} - C γ_{15} - C_{6} γ_{16} > 0, \\ γ_{15} - 2 γ_{16} > 0, \end{array}$ we have $\begin{array}{l} \frac{d}{d t} E_{4} + C_{1} ‖ \nabla Δ w ‖^{2} + C_{2} ‖ \nabla Δ ξ ‖^{2} + C_{3} {‖ \frac{\partial^{2} w}{\partial t^{2}} ‖}_{H^{1}}^{2} + C_{4} {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} \\ (47) & + C_{5} {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{3}}^{2} + C_{6} {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{2}}^{2} ⩽ C_{7} ‖ Δ u ‖^{2} + C_{8} ‖ Δ v ‖^{2}, C_{i} > 0, \end{array}$ where $\begin{array}{l} E_{4} = & γ_{12} (ϵ {‖ \frac{\partial w}{\partial t} ‖}_{H^{2}}^{2} + ‖ w ‖_{H^{3}}^{2}) + γ_{13} {‖ \frac{\partial w}{\partial t} ‖}_{H^{1}}^{2} + γ_{14} (‖ Δ w ‖^{2} + 2 ϵ (Δ \frac{\partial w}{\partial t}, Δ w)) \\ + γ_{15} (‖ ξ ‖_{H^{3}}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{2}}^{2}) + γ_{16} (‖ Δ ξ ‖^{2} + ‖ \nabla Δ ξ ‖^{2} + 2 (Δ \frac{\partial ξ}{\partial t}, Δ ξ)) . \end{array}$ We obtain in particular $\begin{array}{l} (48) & \frac{d}{d t} E_{4} ⩽ C_{7} ‖ Δ u ‖^{2} + C_{8} ‖ Δ v ‖^{2} . \end{array}$ We have $\begin{array}{l} (49) & γ_{12} ϵ {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} + γ_{14} (‖ Δ w ‖^{2} + 2 ϵ (Δ \frac{\partial w}{\partial t}, Δ w)) ⩾ ϵ (γ_{12} - 2 γ_{14}) {‖ Δ \frac{\partial w}{\partial t} ‖}^{2} + \frac{γ_{14}}{2} ‖ Δ w ‖^{2}, \end{array}$ and $\begin{array}{l} (50) & γ_{15} {‖ Δ \frac{\partial ξ}{\partial t} ‖}^{2} + γ_{16} (‖ Δ ξ ‖^{2} + 2 (\nabla \frac{\partial ξ}{\partial t}, Δ ξ)) ⩾ (γ_{15} - 2 γ_{16}) {‖ Δ \frac{\partial ξ}{\partial t} ‖}^{2} + \frac{γ_{16}}{2} ‖ Δ ξ ‖^{2} . \end{array}$ Choosing $γ_{12}$ and $γ_{14}$ such that $\begin{array}{l} γ_{12} - 2 γ_{14} > 0, \end{array}$ there exist a positive constant C independent of ϵ such that $\begin{array}{l} C^{- 1} ({‖ w (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial w (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial w (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ ξ (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ (t)}{\partial t} ‖}_{H^{2}}^{2}) \\ (51) & ⩽ E_{4} (t) ⩽ C ({‖ w (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial w (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial w (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ ξ (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ (t)}{\partial t} ‖}_{H^{2}}^{2}) . \end{array}$ Using the Theorem 3.1 (see [9]) and the relation (33), the estimate (48) becomes $\begin{array}{l} \frac{d}{d t} E_{4} ⩽ Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

By integrating 0 à $t \in [0, T]$ , we have $\begin{array}{l} E_{4} (t) & ⩽ \int_{0}^{t} (Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}) e^{- β τ} + C) d τ \\ (52) & ⩽ (1 + T^{2}) Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}), \end{array}$ where Q a monotonic function are independent of ϵ.

Therefore $\begin{array}{l} (53) & {‖ S_{ϵ}^{2} (t) (ζ_{w} (0), ξ (0), \frac{\partial ξ (0)}{\partial t}) ‖}_{Φ_{2}}^{2} ⩽ (1 + T^{2}) Q (D [u_{0}], {‖ (ζ_{u} (0), α_{0}, α_{1}) ‖}_{Φ_{1}}^{2}), \end{array}$ where Q a monotonic function are independent of ϵ.

The semigroup $S_{ϵ}^{2} (t)$ is regularizing in $Φ_{2}$ .

Using the estimates (41) and (53) there is a global attractor associated with the problem (1)–(4). □

We now have the exponential attractor.
6. Exponential attractor

6.1. Regularization property of the semigroup

Consider two solutions $(u^{(1)}, α^{(1)})$ and $(u^{(2)}, α^{(2)})$ separate from the problem (1)–(4) with the respective initial conditions $(u^{(1)} (0), u_{1}^{(1)}, α^{(1)} (0), α_{1}^{(1)})$ and $(u^{(2)} (0), u_{1}^{(2)}, α^{(2)} (0), α_{1}^{(2)})$ in $B_{R}^{1} (ϵ)$ . Asking $u = u^{(1)} - u^{(2)}$ and $α = α^{(1)} - α^{(2)}$ , $(u, α)$ is solution of the following system $\begin{array}{l} (54) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u + f (u^{(1)}) - f (u^{(2)}) = α, \\ (55) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \end{array}$ which is written in the following form $\begin{array}{l} (56) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u + l (t) u = α, \\ (57) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \\ (58) & u |_{\partial Ω} = α |_{\partial Ω} = 0, \\ (59) & u |_{t = 0} = u (0), \frac{\partial u}{\partial t} |_{t = 0} = u_{1}, α |_{t = 0} = α (0), \frac{\partial α}{\partial t} |_{t = 0} = α_{1} . \end{array}$ The function l is defined by $\begin{array}{l} l (t) = \int_{0}^{1} f^{'} (s u^{(1)} (t) + (1 - s) u^{(2)} (t)) d s, \end{array}$ and there is a positive constant C such that $\begin{array}{l} (60) & for all t \in [0, T], | l (t) | ⩽ C; (see [9]) i.e. ‖ l ‖_{L^{\infty}} ⩽ C . \end{array}$ We break down the operator $S_{ϵ} (t)$ , $t ⩾ 0$ as follows $\begin{array}{l} S_{ϵ} (t) = S_{ϵ}^{1} (t) + S_{ϵ}^{2} (t), \end{array}$ where the operator $S_{ϵ}^{1} (t)$ is defined by: $\begin{array}{l} S_{ϵ}^{1} (t) : Φ_{1} ⟶ Φ_{1}, \\ (ζ_{v^{1}} (0), η^{1} (0), \frac{\partial η^{1} (0)}{\partial t}) ⟼ (ζ_{v^{1}} (t), η^{1} (t), \frac{\partial η^{1} (t)}{\partial t}), \end{array}$ tends to 0 when t goes to infinity, and the operator $S_{ϵ}^{2} (t)$ is defined by: $\begin{array}{l} S_{ϵ}^{2} (t) : Φ_{2} ⟶ Φ_{2}, \\ (ζ_{w^{2}} (0), ξ^{2} (0), \frac{\partial ξ^{2} (0)}{\partial t}) ⟼ (ζ_{w^{2}} (t), ξ^{2} (t), \frac{\partial ξ^{2} (t)}{\partial t}), \end{array}$ is regularizing.

For that $\begin{array}{l} (u, α) = (v^{1}, η^{1}) + (w^{2}, ξ^{2}), \end{array}$ where $(v^{1}, η^{1})$ is solution of the linearity system $\begin{array}{l} (61) & ϵ \frac{\partial^{2} v^{1}}{\partial t^{2}} + \frac{\partial v^{1}}{\partial t} - Δ v^{1} = η^{1}, \\ (62) & \frac{\partial^{2} η^{1}}{\partial t^{2}} + \frac{\partial η^{1}}{\partial t} - Δ \frac{\partial η^{1}}{\partial t} - Δ η^{1} = - v^{1} - \frac{\partial v^{1}}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{l} (63) & ζ_{v^{1}} |_{\partial Ω} = η^{1} |_{\partial Ω} = 0, \end{array}$ and initial conditions $\begin{array}{l} (64) & ζ_{v^{1}} |_{t = 0} = ζ_{v^{1}} (0), η^{1} |_{t = 0} = α (0), \frac{\partial η^{1}}{\partial t} |_{t = 0} = α_{1}, \end{array}$ and $(w^{2}, ξ^{2})$ is solution of the nonlinearity system $\begin{array}{l} (65) & ϵ \frac{\partial^{2} w^{2}}{\partial t^{2}} + \frac{\partial w^{2}}{\partial t} - Δ w^{2} + l (t) u = ξ^{2}, \\ (66) & \frac{\partial^{2} ξ^{2}}{\partial t^{2}} + \frac{\partial ξ^{2}}{\partial t} - Δ \frac{\partial ξ^{2}}{\partial t} - Δ ξ^{2} = - w^{2} - \frac{\partial w^{2}}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{l} (67) & ζ_{w^{2}} |_{\partial Ω} = ξ^{2} |_{\partial Ω} = 0, \end{array}$ and initial conditions $\begin{array}{l} (68) & ζ_{w^{2}} |_{t = 0} = ξ^{2} |_{t = 0} = \frac{\partial ξ^{2}}{\partial t} |_{t = 0} = 0 . \end{array}$

Lemma 6.1.
Let $ϵ ⩽ 1$ , and $(v^{1}, η^{1})$ the solution of the problem ( 61 )–( 64 ), with initial conditions in $Φ_{1}$ , then $(v^{1}, η^{1})$ satisfies $\begin{array}{l} {‖ S_{ϵ}^{1} (t) (ζ_{v^{1}} (0), η^{1} (0), \frac{\partial η^{1} (0)}{\partial t}) ‖}_{Φ_{1}}^{2} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t}, \end{array}$

where β positive constant and Q a monotonic function are independent of ϵ.
Proof.
Multiply (61) by $- Δ \frac{\partial v^{1}}{\partial t}$ and integrate over Ω, we obtain $\begin{array}{l} (69) & \frac{d}{d t} (ϵ {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + {‖ Δ v^{1} ‖}^{2}) + {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} ⩽ 2 c_{p} {‖ Δ η^{1} ‖}^{2} . \end{array}$ Multiplying (61) by $- Δ v^{1}$ , then integrating over Ω and applying Hölder and Young inequalities, we find $\begin{array}{l} (70) & \frac{d}{d t} ({‖ \nabla v^{1} ‖}^{2} + 2 ϵ (\nabla \frac{\partial v^{1}}{\partial t}, \nabla v^{1})) + {‖ Δ v^{1} ‖}^{2} ⩽ 2 {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + c_{p}^{2} {‖ Δ η^{1} ‖}^{2} . \end{array}$ Multiplying (62) by $- Δ η^{1}$ and integrating over Ω, using Hölder and Young inequalities, we obtain the following: $\begin{array}{l} \frac{d}{d t} ({‖ \nabla η^{1} ‖}^{2} + {‖ Δ η^{1} ‖}^{2} + 2 (\nabla \frac{\partial η^{1}}{\partial t}, \nabla η^{1})) + {‖ Δ η^{1} ‖}^{2} \\ (71) & ⩽ 2 {‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + 2 c_{p}^{2} {‖ Δ v^{1} ‖}^{2} + 2 c_{p} {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} . \end{array}$ Multiplying (62) by $- Δ \frac{\partial η^{1}}{\partial t}$ , then we integrate over Ω and applying Hölder and Young inequalities, we obtain $\begin{array}{l} (72) & \frac{d}{d t} ({‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + {‖ Δ η^{1} ‖}^{2}) + {‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial η^{1}}{\partial t} ‖}^{2} ⩽ c_{p}^{2} {‖ Δ v^{1} ‖}^{2} + {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{1}$ (69), $γ_{2}$ (70), $γ_{3}$ (71) and $γ_{4}$ (72) with $γ_{1}$ , $γ_{2}$ , $γ_{3}$ and $γ_{4} > 0$ , such that $\begin{array}{l} γ_{1} - 2 γ_{2} - 2 c_{p} γ_{3} - γ_{4} > 0, \\ γ_{2} - 2 c_{p}^{2} γ_{3} - c_{p}^{2} γ_{4} > 0, \\ γ_{3} - 2 c_{p} γ_{1} - c_{p}^{2} γ_{2} > 0, \end{array}$ we have $\begin{array}{l} (73) & \frac{d}{d t} E_{5} + C_{1} {‖ Δ η^{1} ‖}^{2} + C_{2} {‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + C_{3} {‖ Δ v^{1} ‖}^{2} + C_{4} {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + C_{5} {‖ Δ \frac{\partial η^{1}}{\partial t} ‖}^{2} ⩽ 0, \end{array}$ where $\begin{array}{l} E_{5} = & γ_{1} (ϵ {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + {‖ Δ v^{1} ‖}^{2}) + γ_{2} ({‖ \nabla v^{1} ‖}^{2} + 2 ϵ (\nabla \frac{\partial v^{1}}{\partial t}, \nabla v^{1})) \\ + γ_{3} ({‖ \nabla η^{1} ‖}^{2} + {‖ Δ η^{1} ‖}^{2} + 2 (\nabla \frac{\partial η^{1}}{\partial t}, \nabla η^{1})) + γ_{4} ({‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + {‖ Δ η^{1} ‖}^{2}) . \end{array}$

We have $\begin{array}{l} (74) & γ_{1} ϵ {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + γ_{2} ({‖ \nabla v^{1} ‖}^{2} + 2 ϵ (\nabla \frac{\partial v^{1}}{\partial t}, \nabla v^{1})) ⩾ ϵ (γ_{1} - 2 γ_{2}) {‖ \nabla \frac{\partial v^{1}}{\partial t} ‖}^{2} + \frac{γ_{2}}{2} {‖ \nabla v^{1} ‖}^{2}, \end{array}$ and $\begin{array}{l} (75) & γ_{4} {‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + γ_{3} ({‖ \nabla η^{1} ‖}^{2} + 2 (\nabla \frac{\partial η^{1}}{\partial t}, \nabla η^{1})) ⩾ (γ_{4} - 2 γ_{3}) {‖ \nabla \frac{\partial η^{1}}{\partial t} ‖}^{2} + \frac{γ_{3}}{2} {‖ \nabla η^{1} ‖}^{2} . \end{array}$ Choosing $γ_{1}$ , $γ_{2}$ , $γ_{3}$ and $γ_{4}$ such that $\begin{array}{l} γ_{1} - 2 γ_{2} > 0, \\ γ_{4} - 2 γ_{3} > 0, \end{array}$ there exist a positive constant C independent of ϵ such that $\begin{array}{l} C^{- 1} ({‖ v^{1} (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v^{1} (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η^{1} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η^{1} (t)}{\partial t} ‖}_{H^{1}}^{2}) \\ (76) & ⩽ E_{5} (t) ⩽ C ({‖ v^{1} (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v^{1} (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η^{1} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η^{1} (t)}{\partial t} ‖}_{H^{1}}^{2}) . \end{array}$ Using the estimate (76), we result in then with an inequality of the following form $\begin{array}{l} (77) & \frac{d}{d t} E_{5} + β E_{5} + C_{1} {‖ \frac{\partial v^{1}}{\partial t} ‖}_{H^{1}}^{2} + C_{2} {‖ \frac{\partial η^{1}}{\partial t} ‖}_{H^{1}}^{2} ⩽ 0, \end{array}$ where β and $C_{i}$ are positive constants independent of ϵ.

Gronwall’s lemma gives, for all $t \in [0, T]$ $\begin{array}{l} (78) & E_{5} (t) + \int_{0}^{t} ({‖ \frac{\partial v^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial η^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ ⩽ E_{5} (0) e^{- β t} . \end{array}$ Using the estimate (76), we find $\begin{array}{l} {‖ v^{1} (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial v^{1} (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ η^{1} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial η^{1} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (79) & + \int_{0}^{t} ({‖ \frac{\partial v^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial η^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t} \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

Let us consider the following equation $\begin{array}{l} (80) & ϵ \frac{\partial^{2} v^{1}}{\partial t^{2}} + \frac{\partial v^{1}}{\partial t} - Δ v^{1} = h (t) = η^{1}, \\ (81) & ζ_{v^{1}} (t) |_{\partial Ω} = h (t) |_{\partial Ω} = 0 . \end{array}$ The application of the Proposition A.2 (see [6]) to equation (80) gives $\begin{array}{l} {‖ ζ_{v^{1}} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial v^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ C ({‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ h (0) ‖}^{2}) e^{- β t} \\ (82) & + C \int_{0}^{t} ({‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2}) e^{- β (t - τ)} d τ . \end{array}$ Let us determine an estimate of $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ We have $\begin{array}{l} (83) & {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} = {‖ η^{1} (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial η^{1} (τ)}{\partial t} ‖}_{H^{- 1}}^{2} \end{array}$ Let us determine the estimates $‖ η^{1} (τ) ‖_{H^{1}}$ and $‖ \frac{\partial η^{1} (τ)}{\partial t} ‖_{H^{- 1}}^{2}$ .

Using the estimate (79), we have $\begin{array}{l} (84) & {‖ η^{1} (τ) ‖}_{H^{1}} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{\frac{- β τ}{2}} . \end{array}$ Then $\begin{array}{l} (85) & {‖ \frac{\partial η^{1} (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{\frac{- β τ}{2}} . \end{array}$ The estimate becomes $\begin{array}{l} (86) & {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β τ} . \end{array}$ Using (86) and (79), the estimate (82) becomes, for all $t \in [0, T]$ $\begin{array}{l} (87) & {‖ ζ_{v^{1}} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial v^{1} (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t} . \end{array}$ Using the estimates (79) and (87), we obtain $\begin{array}{l} {‖ (ζ_{v^{1}} (t), η^{1} (t), \frac{\partial η^{1} (t)}{\partial t}) ‖}_{Φ_{1}}^{2} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t}, \end{array}$ therefore $\begin{array}{l} {‖ S_{ϵ}^{1} (t) (ζ_{v^{1}} (0), η^{1} (0), \frac{\partial η^{1} (0)}{\partial t}) ‖}_{Φ_{1}}^{2} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t}, \end{array}$ where β a positive constant and Q a monotonic function are independent of ϵ. □

We now consider (65)–(68).

We state the following lemma. Lemma 6.2.
Let $ϵ ⩽ 1$ , and $(w^{2}, ξ^{2})$ the solution of the problem ( 65 )–( 68 ), with initiales conditions in $B_{R_{1}} (ϵ) \cap Φ_{2}$ , then $(w^{2}, ξ^{2})$ satisfies the following estimate $\begin{array}{l} {‖ S_{ϵ}^{2} (t) (ζ_{w^{2}} (0), ξ^{2} (0), \frac{\partial ξ^{2} (0)}{\partial t}) ‖}_{Φ_{2}}^{2} ⩽ C e^{k t} Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}), \end{array}$ where C and k the positive constants are independent of ϵ.
Proof.
Multiplying (65) by $Δ^{2} \frac{\partial w^{2}}{\partial t}$ and we integrate over Ω, we have $\begin{array}{l} \frac{d}{d t} (ϵ {‖ \frac{\partial w^{2}}{\partial t} ‖}_{H^{2}}^{2} + {‖ w^{2} ‖}_{H^{3}}^{2}) + 2 {‖ \frac{\partial w^{2}}{\partial t} ‖}_{H^{2}}^{2} \\ ⩽ 2 (‖ l (t) Δ u ‖ + ‖ Δ l (t) . u ‖ + 2 ‖ \nabla u \nabla l (t) ‖) ‖ Δ \frac{\partial w^{2}}{\partial t} ‖ + 2 ‖ Δ \frac{\partial w^{2}}{\partial t} ‖ ‖ Δ ξ^{2} ‖ . \end{array}$ We know that $\begin{array}{l} ‖ \nabla l (t) ‖ & = ‖ \nabla \int_{0}^{1} f^{'} (s u^{(1)} + (1 - s) u^{(2)}) d s ‖ \\ ⩽ ‖ f^{″} (s u^{(1)} + (1 - s) u^{(2)}) ‖ (‖ \nabla u^{(1)} + \nabla u^{(2)} ‖) \\ (88) & ⩽ C_{1}, C_{1} > 0, \end{array}$ and $\begin{array}{l} ‖ Δ l (t) ‖ = & ‖ \nabla (\int_{0}^{1} f^{″} (s u^{(1)} + (1 - s) u^{(2)}) \nabla (s u^{(1)} + (1 - s) u^{(2)})) d s ‖ \\ ⩽ & \int_{0}^{1} (‖ f^{‴} (s u^{(1)} + (1 - s) u^{(2)}) ‖ {(‖ \nabla (u^{(1)} + u^{(2)}) ‖)}^{2} \\ + ‖ f^{″} (s u^{(1)} + (1 - s) u^{(2)}) ‖ ‖ Δ (u^{(1)} + u^{(2)}) ‖) d s \\ (89) & ⩽ & C_{2}, C_{2} > 0 . \end{array}$ We obtain, by inserting the estimates (88), (89) and using the Hölder and Young inequalities: $\begin{array}{l} \frac{d}{d t} (ϵ {‖ \frac{\partial w^{2}}{\partial t} ‖}_{H^{2}}^{2} + {‖ w^{2} ‖}_{H^{3}}^{2}) + {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} \\ (90) & ⩽ C_{1} {‖ w^{2} ‖}_{H^{2}}^{2} + C_{1} {‖ v^{1} ‖}_{H^{2}}^{2} + C_{2} {‖ \nabla Δ ξ^{2} ‖}^{2}, C_{i} > 0 . \end{array}$ We multiply (65) by $Δ^{2} w^{2}$ and we integrate over Ω, we have $\begin{array}{l} \frac{d}{d t} ({‖ Δ w^{2} ‖}^{2} + 2 ϵ (Δ \frac{\partial w^{2}}{\partial t}, Δ w^{2})) + {‖ w^{2} ‖}_{H^{3}}^{2} \\ (91) & ⩽ 2 {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} + C_{3} c_{p} {‖ Δ ξ^{2} ‖}^{2} + C_{4} {‖ w^{2} ‖}_{H^{2}}^{2} + C_{4} {‖ v^{1} ‖}_{H^{2}}^{2}, C_{i} > 0 . \end{array}$

Multiplying (65) by $- Δ \frac{\partial^{2} w^{2}}{\partial t^{2}}$ and we integrate over Ω, using the Hölder and Young inequalities: $\begin{array}{l} (92) & \frac{d}{d t} {‖ \nabla \frac{\partial w^{2}}{\partial t} ‖}^{2} + ϵ {‖ \frac{\partial^{2} w^{2}}{\partial t^{2}} ‖}_{H^{1}}^{2} ⩽ C_{5} {‖ \nabla Δ w^{2} ‖}^{2} + C_{6} {‖ Δ v^{1} ‖}^{2} + C_{7} {‖ Δ \nabla ξ^{2} ‖}^{2} . \end{array}$ Multiplying the relation (66) by $Δ^{2} \frac{\partial ξ^{2}}{\partial t}$ and integrate over Ω, we obtain using the Hölder and Young inequalities $\begin{array}{l} (93) & \frac{d}{d t} ({‖ ξ^{2} ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ^{2}}{\partial t} ‖}_{H^{2}}^{2}) + 2 {‖ \frac{\partial ξ^{2}}{\partial t} ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ^{2}}{\partial t} ‖}_{H^{2}}^{2} ⩽ 2 {‖ Δ w^{2} ‖}^{2} + 2 {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} . \end{array}$ Multiplying (66) by $Δ^{2} ξ^{2}$ and we integrate over Ω, we have $\begin{array}{l} \frac{d}{d t} ({‖ Δ ξ^{2} ‖}^{2} + {‖ \nabla Δ ξ^{2} ‖}^{2} + 2 (Δ \frac{\partial ξ^{2}}{\partial t}, Δ ξ^{2})) + {‖ \nabla Δ ξ^{2} ‖}^{2} \\ (94) & ⩽ 2 {‖ Δ \frac{\partial ξ^{2}}{\partial t} ‖}^{2} + C_{8} {‖ \nabla Δ w^{2} ‖}^{2} + C_{9} {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{13}$ (90), $γ_{14}$ (91), $γ_{15}$ (92), $γ_{16}$ (93) and $γ_{17}$ (94), with $γ_{13}$ , $γ_{14}$ , $γ_{15}$ , $γ_{16}$ and $γ_{17} > 0$ , such that $\begin{array}{l} γ_{13} - 2 γ_{14} - 2 γ_{16} - C_{9} γ_{17} > 0, \\ γ_{14} - C_{5} γ_{15} - C_{8} γ_{17} > 0, \\ γ_{17} - C_{2} γ_{13} - C_{7} γ_{15} > 0, \end{array}$ we have $\begin{array}{l} \frac{d}{d t} E_{6} + C_{1} {‖ \nabla Δ w^{2} ‖}^{2} + C_{2} {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} + C_{3} {‖ \nabla Δ ξ^{2} ‖}^{2} + C_{4} {‖ Δ \frac{\partial ξ^{2}}{\partial t} ‖}^{2} + C_{5} {‖ \nabla Δ \frac{\partial ξ^{2}}{\partial t} ‖}^{2} \\ (95) & ⩽ C_{6} {‖ Δ w^{2} ‖}^{2} + C_{7} {‖ Δ v^{1} ‖}^{2}, \end{array}$ where the constants $C_{i}$ are positive and independent of ϵ, and $\begin{array}{l} E_{6} = & γ_{13} (ϵ {‖ \frac{\partial w^{2}}{\partial t} ‖}_{H^{2}}^{2} + {‖ w^{2} ‖}_{H^{3}}^{2}) + γ_{14} ({‖ Δ w^{2} ‖}^{2} + 2 ϵ (Δ \frac{\partial w^{2}}{\partial t}, Δ w^{2})) + γ_{15} {‖ \frac{\partial w^{2}}{\partial t} ‖}_{H^{1}}^{2} \\ + γ_{16} ({‖ ξ^{2} ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ^{2}}{\partial t} ‖}_{H^{2}}^{2}) + γ_{17} ({‖ Δ ξ^{2} ‖}^{2} + {‖ \nabla Δ ξ^{2} ‖}^{2} + 2 (Δ \frac{\partial ξ^{2}}{\partial t}, Δ ξ^{2})) . \end{array}$ We know that $\begin{array}{l} γ_{13} ϵ {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} + γ_{14} ({‖ Δ w^{2} ‖}^{2} + 2 ϵ (Δ \frac{\partial w^{2}}{\partial t}, Δ w^{2})) \\ (96) & ⩾ ϵ (γ_{13} - 2 γ_{14}) {‖ Δ \frac{\partial w^{2}}{\partial t} ‖}^{2} + \frac{γ_{14}}{2} {‖ Δ w^{2} ‖}^{2}, \end{array}$ and $\begin{array}{l} (97) & γ_{16} {‖ Δ \frac{\partial ξ^{2}}{\partial t} ‖}^{2} + γ_{17} ({‖ Δ ξ^{2} ‖}^{2} + 2 (\nabla \frac{\partial ξ^{2}}{\partial t}, Δ ξ^{2})) ⩾ (γ_{16} - 2 γ_{17}) {‖ Δ \frac{\partial ξ^{2}}{\partial t} ‖}^{2} + \frac{γ_{17}}{2} {‖ Δ ξ^{2} ‖}^{2} . \end{array}$ Choosing $γ_{13}$ , $γ_{14}$ , $γ_{16}$ and $γ_{17}$ such that $\begin{array}{l} γ_{13} - 2 γ_{14} > 0, \\ γ_{16} - 2 γ_{17} > 0, \end{array}$ there exist a positive constant C independent of ϵ such that $\begin{array}{l} C^{- 1} ({‖ w^{2} (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ ξ^{2} (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ^{2} (t)}{\partial t} ‖}_{H^{2}}^{2}) \\ (98) & ⩽ E_{6} (t) ⩽ C ({‖ w^{2} (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ ξ^{2} (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial ξ^{2} (t)}{\partial t} ‖}_{H^{2}}^{2}) . \end{array}$ We obtain using the relation (98) $\begin{array}{l} (99) & \frac{d}{d t} E_{6} ⩽ k E_{6} + C {‖ Δ v^{1} ‖}^{2}, C > 0 . \end{array}$ Using the estimate (79), we find $\begin{array}{l} \frac{d}{d t} E_{6} - k E_{6} ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) e^{- β t} . \end{array}$ Gronwall’s lemma gives $\begin{array}{l} E_{6} (t) e^{- k t} & ⩽ Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) \int_{0}^{t} e^{- (β + k) τ} d τ \\ ⩽ C e^{k t} Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) . \end{array}$

Using (98), we obtain the following estimate: $\begin{array}{l} {‖ w^{2} (t) ‖}_{H^{3}}^{2} + {‖ ξ^{2} (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial w^{2} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ + {‖ \frac{\partial ξ^{2} (t)}{\partial t} ‖}_{H^{2}}^{2} ⩽ C e^{k t} Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}) . \end{array}$ Then $\begin{array}{l} {‖ (ζ_{w^{2}} (t), ξ^{2} (t), \frac{\partial ξ^{2} (t)}{\partial t}) ‖}_{Φ_{2}}^{2} ⩽ C e^{k t} Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}), \end{array}$ therefore $\begin{array}{l} (100) & {‖ S_{ϵ}^{2} (t) (ζ_{w^{2}} (0), ξ^{2} (0), \frac{\partial ξ^{2} (0)}{\partial t}) ‖}_{Φ_{2}}^{2} ⩽ C e^{k t} Q ({‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}), \end{array}$ with C and k are positive constants and Q a monotonic function are independent of ϵ. □

6.2. Hölder continuous in time

We will then show that the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , is Hölder continuous with respect to $t \in [0, T]$ .

We state the following lemma

Lemma 6.3.
We assume verified the hypotheses of Theorem 5.1 and $(ζ_{u} (0), α (0), α_{1}) \in Φ_{1}$ , then there is a positive constant C(T) such that for all $t_{1}, t_{2} \in [0, T]$ , where $t_{1} ⩽ t_{2}$ $\begin{array}{l} {‖ S_{ϵ} (t_{1}) (ζ_{u} (0), α (0), α_{1}) - S_{ϵ} (t_{2}) (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}} ⩽ C (T) | t_{2} - t_{1} |^{\frac{1}{2}} . \end{array}$
Proof.
Using the Hölder inequality, we have $\begin{array}{l} {‖ S_{ϵ} (t_{1}) (ζ_{u} (0), α (0), α_{1}) - S_{ϵ} (t_{2}) (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2} \\ ⩽ | t_{2} - t_{1} | (\int_{t_{1}}^{t_{2}} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} d τ + \int_{t_{1}}^{t_{2}} ϵ {‖ \frac{\partial^{2} u (τ)}{\partial t^{2}} ‖}_{H^{1}}^{2} d τ + \int_{t_{1}}^{t_{2}} {‖ \frac{\partial^{2} u (τ)}{\partial t^{2}} ‖}^{2} d τ \\ + \int_{t_{1}}^{t_{2}} {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2} d τ + \int_{t_{1}}^{t_{2}} {‖ \frac{\partial^{2} α (τ)}{\partial t^{2}} ‖}_{H^{1}}^{2} d τ) . \end{array}$ We examine $\int_{t_{1}}^{t_{2}} ‖ \frac{\partial^{2} u (τ)}{\partial t^{2}} ‖_{H^{1}}^{2} d τ$ and $\int_{t_{1}}^{t_{2}} ‖ \frac{\partial^{2} α (τ)}{\partial t^{2}} ‖_{H^{1}}^{2} d τ$ .

Multiplying (1) by $- 2 Δ \frac{\partial^{2} u}{\partial t^{2}}$ and integrating over Ω, we have applying Hölder and Young inequalities $\begin{array}{l} (101) & \frac{d}{d t} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + \frac{ϵ}{2} {‖ \nabla \frac{\partial^{2} u}{\partial t^{2}} ‖}^{2} ⩽ C, \end{array}$ in particular $\begin{array}{l} (102) & {‖ \nabla \frac{\partial^{2} u}{\partial t^{2}} ‖}^{2} ⩽ C, \end{array}$ after the integration of (102) between $t_{1}$ and $t_{2}$ , we obtain $\begin{array}{l} (103) & \int_{t_{1}}^{t_{2}} {‖ \frac{\partial^{2} u (τ)}{\partial t^{2}} ‖}_{H^{1}}^{2} d τ ⩽ C . \end{array}$ Multiplying (2) by $- 2 Δ \frac{\partial^{2} α}{\partial t^{2}}$ and integrate over Ω, we have using Hölder and Young inequalities $\begin{array}{l} (104) & \frac{d}{d t} ({‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial α}{\partial t} ‖}^{2}) + \frac{1}{2} {‖ \nabla \frac{\partial^{2} α}{\partial t^{2}} ‖}^{2} ⩽ C, \end{array}$ in particular $\begin{array}{l} (105) & {‖ \nabla \frac{\partial^{2} α}{\partial t^{2}} ‖}^{2} ⩽ C, \end{array}$ after the integration of (105) between $t_{1}$ and $t_{2}$ , we obtain $\begin{array}{l} (106) & \int_{t_{1}}^{t_{2}} {‖ \frac{\partial^{2} α (τ)}{\partial t^{2}} ‖}_{H^{1}}^{2} d τ ⩽ C . \end{array}$ The estimate (103) and (106) implies $\begin{array}{l} (107) & {‖ S_{ϵ} (t_{1}) (ζ_{u} (0), α (0), α_{1}) - S_{ϵ} (t_{2}) (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}} ⩽ C (T) | t_{2} - t_{1} |^{\frac{1}{2}} . \end{array}$ □

6.3. Continuous Lipschitz

Finally, we show that the semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , is Lipschitzian with respect to space.

Lemma 6.4.

Assume the hypothesis of Theorem 5.1 .

If ${(ζ_{u}, α, \frac{\partial α}{\partial t}) \in Φ_{1} / ‖ (ζ_{u}, α, \frac{\partial α}{\partial t}) ‖_{Φ_{1}}^{2} ⩽ R_{1}^{2}}$ is a bounded subset of $Φ_{1}$ , then there exist a constant $C_{R_{1}} (t)$ such that for all $(ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) a n d (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)}) \in Φ_{1}$ with $‖ (ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) ‖_{Φ_{1}}^{2} ⩽ R_{1}^{2}$ and $‖ (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)}) ‖_{Φ_{1}}^{2} ⩽ R_{1}^{2}$ , we have $\begin{array}{l} {‖ S_{ϵ} (t) (ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) - S_{ϵ} (t) (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} \\ ⩽ C_{R_{1}} (t) {‖ (ζ_{u^{(1)}} (0) - ζ_{u^{(2)}} (0), α^{(1)} (0) - α^{(2)} (0), α_{1}^{(1)} - α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} . \end{array}$

$(u^{(1)}, α^{(1)})$ and $(u^{(2)}, α^{(2)})$ are the solutions of problem (1)–(4) with respectively the initial conditions $(ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) and (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)})$ .

Proof.

Indeed $\begin{array}{l} {‖ S_{ϵ} (t) (ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) - S_{ϵ} (t) (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} \\ (108) & = {‖ (ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖}_{Φ_{1}}^{2} . \end{array}$ Multiplying (56) by $- Δ \frac{\partial u}{\partial t}$ and integrating over Ω, then using the Hölder and Young inégalities, we arrive to $\begin{array}{l} (109) & \frac{d}{d t} (‖ Δ u ‖^{2} + ϵ {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ ‖ \nabla α ‖^{2} + C ‖ \nabla u ‖^{2} . \end{array}$ Multiplying (57) by $- Δ \frac{\partial α}{\partial t}$ and integrating over Ω, we have using the Hölder and Young inégalities $\begin{array}{l} \frac{d}{d t} (‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + 2 {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} ⩽ 2 ‖ \nabla u ‖^{2} + 2 {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}, \end{array}$ in particular $\begin{array}{l} (110) & \frac{d}{d t} (‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2}) + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} ⩽ 2 ‖ \nabla u ‖^{2} + 2 {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$ Adding (109) and $γ_{18}$ (110), with $γ_{18} > 0$ such that $1 - 2 γ_{18} > 0$ , we arrive to $\begin{array}{l} (111) & \frac{d}{d t} E_{7} + C {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} ⩽ C ‖ \nabla u ‖^{2} + ‖ \nabla α ‖^{2} ⩽ C (R_{1}^{2} + 1) E_{7}, \end{array}$ where $\begin{array}{l} E_{7} = ‖ Δ u ‖^{2} + ϵ {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + γ_{18} (‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2}) . \end{array}$ Gronwall’s lemma gives $\begin{array}{l} E_{7} (t) + \int_{0}^{t} ({‖ \nabla \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α (τ)}{\partial t} ‖}^{2}) d τ ⩽ ({‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{C (R_{1}^{2} + 1) t}, \end{array}$ therefore $\begin{array}{l} {‖ Δ u (t) ‖}^{2} + ϵ {‖ \nabla \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ Δ α (t) ‖}^{2} + {‖ \nabla \frac{\partial α (t)}{\partial t} ‖}^{2} \\ (112) & + \int_{0}^{t} ({‖ \nabla \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α (τ)}{\partial t} ‖}^{2}) d τ ⩽ C {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2} e^{C (R_{1}^{2} + 1) t} . \end{array}$

We consider the following hyperbolic equation $\begin{array}{l} (113) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u = h (t) = - f (u^{(1)}) + f (u^{(2)}) + α, \\ (114) & ζ_{u} (t) |_{\partial Ω} = h (t) |_{\partial Ω} = 0 . \end{array}$ The application of the Proposition A.2 (see [6]) to equation (113) gives $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ (115) & ⩽ C ({‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ h (0) ‖}^{2}) e^{- β t} + C \int_{0}^{t} ({‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2}) e^{- β (t - τ)} d τ . \end{array}$ Let us determine an estimate of $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ We have $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} \\ ⩽ {‖ α (τ) ‖}_{H^{1}}^{2} + {‖ f (u^{(1)} (τ)) - f (u^{(2)} (τ)) ‖}_{H^{1}}^{2} + {‖ f^{'} (u^{(1)} (τ)) \frac{\partial u (τ)}{\partial t} ‖}_{H^{- 1}}^{2} \\ (116) & + {‖ f^{'} (u^{(1)} (τ)) - f^{'} (u^{(2)} (τ)) \frac{\partial u^{(2)} (τ)}{\partial t} ‖}_{H^{- 1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ Let us determine the estimates: $‖ α (τ) ‖_{H^{1}}$ , $‖ f (u^{(1)} (τ)) - f (u^{(2)} (τ)) ‖_{H^{1}}$ , $‖ \frac{\partial α (τ)}{\partial t} ‖_{H^{- 1}}^{2}$ and $‖ f^{'} (u^{(1)} (τ)) - f^{'} (u^{(2)} (τ)) \frac{\partial u^{(2)} (τ)}{\partial t} ‖_{H^{- 1}}$ .

Using the uniforms estimates of $‖ u ‖_{H^{2}}^{2}$ , $‖ α ‖_{H^{2}}^{2}$ and $‖ \frac{\partial α}{\partial t} ‖_{H^{1}}^{2}$ independent of ϵ, we have $\begin{array}{l} (117) & {‖ α (τ) ‖}_{H^{1}} ⩽ C ‖ Δ α (τ) ‖, \end{array}$ and $\begin{array}{l} (118) & {‖ f (u^{(1)} (τ)) - f (u^{(2)} (τ)) ‖}_{H^{1}} ⩽ C ‖ Δ u (τ) ‖, \end{array}$ then, $\begin{array}{l} (119) & {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ C {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}} . \end{array}$ We know that, for all $w \in H_{0}^{1} (Ω)$ , $\begin{array}{l} (120) & | (f^{'} (u^{(1)} (τ)) \frac{\partial u (τ)}{\partial t}, w) | ⩽ C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}} ‖ w ‖_{H^{1}}, \end{array}$ then $\begin{array}{l} (121) & ‖ {(f^{'} (u^{(1)} (τ)) \frac{\partial u (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}} . \end{array}$ By analogy, for all $w \in H_{0}^{1} (Ω)$ , $\begin{array}{l} (122) & | (f^{'} (u^{(1)} (τ)) - f^{'} (u^{(2)} (τ)) \frac{\partial u^{(2)} (τ)}{\partial t}, w) | ⩽ C {‖ \frac{\partial u^{(2)} (τ)}{\partial t} ‖}_{H^{1}} ‖ w ‖_{H^{1}}, \end{array}$ therefore $\begin{array}{l} (123) & ‖ {(f^{'} (u^{(1)} (τ)) - f^{'} (u^{(2)} (τ)) \frac{\partial u^{(2)} (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}} . \end{array}$ The estimate becomes $\begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} ⩽ C ({‖ Δ u (τ) ‖}^{2} + {‖ \nabla \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ Δ α (τ) ‖}^{2} + {‖ \nabla \frac{\partial α (τ)}{\partial t} ‖}^{2}) . \end{array}$ The relation (115) becomes $\begin{array}{l} (124) & {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ ⩽ C {‖ (ζ_{u} (0), α (0), α_{1}) ‖}_{Φ_{1}}^{2} e^{C (R_{1}^{2} + 1) t} . \end{array}$ By combining the estimates (112) and (124), we obtain, for all $t \in [0, T]$ $\begin{array}{l} ‖ ζ_{u} (t), α (t), \frac{\partial α (t)}{\partial t}) ‖_{Φ_{1}}^{2} \\ (125) & ⩽ C_{R_{1}} (t) {‖ (ζ_{u^{(1)}} (0) - ζ_{u^{(2)}} (0), α^{(1)} (0) - α^{(2)} (0), α_{1}^{(1)} - α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} . \end{array}$ Finally $\begin{array}{l} {‖ S_{ϵ} (t) (ζ_{u^{(1)}} (0), α^{(1)} (0), α_{1}^{(1)}) - S_{ϵ} (t) (ζ_{u^{(2)}} (0), α^{(2)} (0), α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} \\ (126) & ⩽ C_{R_{1}} (t) {‖ (ζ_{u^{(1)}} (0) - ζ_{u^{(2)}} (0), α^{(1)} (0) - α^{(2)} (0), α_{1}^{(1)} - α_{1}^{(2)}) ‖}_{Φ_{1}}^{2} . \end{array}$ □

The Lemmas 6.1, 6.2, 6.3 and 6.4 established above authorize us to formulate the theorem entirely shown hereafter by these results.

Theorem 6.1.

The semigroup $S_{ϵ} (t)$ , $t ⩾ 0$ , associated on the problem ( 1 )–( 4 ) possesses an exponential attractor.

Conclusion 6.1.

An exponential attractor being of finite fractal dimension, the global attractor which is contained in an exponential attractor is of finite fractal dimension. Therefore the dynamics of the system can be described by a finite number of degrees of freedom.

Footnotes

Acknowledgements

The authors would like to thank Alain MIRANVILLE for his collaboration and anonymous referees for their valuable comments.

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