Global attractor for a hyperbolic Cahn–Hilliard equation with a proliferation term

Abstract

In this article, we study a hyperbolic equation of Cahn–Hilliard with a proliferation term and Dirichlet boundary conditions. In particular, we prove the existence and uniqueness of the solution, and also the existence of the global attractor.

Keywords

Cahn–Hilliard equation proliferation term dissipativity global attractor comparison principle

1. Introduction

The global attractor is a smallest compact set, invariant by an evolution operator called semigroup associated to an equation or a system, which attracts all bounded sets of initial data when time becomes infinity.

The parabolic Cahn–Hilliard equation is on the following form $\begin{matrix} (1.1) & \frac{\partial u}{\partial t} + Δ^{2} u - Δ f (u) + g (u) = 0, \end{matrix}$ where u is the order parameter, f is the derivative of a double-well potential F or the local free energy and g is the proliferation term, has been proposed in [3,6] as a model for the growth of cancerous tumors and the biological entities.

Here g can be the linear function $g (s) = α s$ , $α > 0$ , in that case, (1.1) is known as the Cahn–Hilliard Oono equation and accounts for long-ranged interactions in the phase equation and in the phase separation process [8].

A second possibility is the nonlinear function $g (s) = α s (s - 1)$ ; $α > 0$ is the proliferation rate. In that case, (1.1) has applications in biology and precisely, models wound healing and tumour growth (in one space dimension) and the clustering of brain tumor cells for other quadratic functions with chemical applications and for other polynomials with biological applications [13].

A third possibility is the nonlinear function $g (s) = α s + β s^{2} {(s - 1)}^{2}$ , $α, β > 0$ , this model describes the tumorous growth uncontrolled of abnormal cells, what often results by a cancer (see [1]); in this case, α and β are two constants who represent the coefficient of death and of the growth, respectively. Note that α and β parameters can also depend of the spatial variable and the time.

A fourth possibility is the function $g (s) = λ_{0} χ_{Ω ∖ D} (x) (s - h (x))$ , $λ_{0} > 0$ , $D \subset Ω$ , $h \in L^{2} (Ω)$ . This possibility was proposed in [2] in view of applications to image inpainting, h is a given image and D is the inpainting (i.e..,damaged) region, χ denotes the indicator function.

The equation (1.1) has been studied in the references [2,3,6,8,10] and [9], in which authors have proved the existence and the uniqueness of the solution and the existence of the finite-dimensional attractors.

Some studies have been done in the same way, but with only a double-well potential acting in a phase transition phenomenon concerned by a parabolic system or hyperbolic or even parabolic-hyperbolic system (see [4,5,7–9,11,12]).

In this paper, we are going to study the model of Cahn–Hilliard type for the growth of cancerous tumors and other biological applications expressed by the following hyperbolic pertubed Cahn–Hilliard equation with a proliferation term $\begin{matrix} (1.2) & ε \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} + Δ^{2} u - Δ f (u) + g (u) = 0, \end{matrix}$ where the proliferation term is a polynomial of any degree, in a smooth and bounded domain $Ω \subset R^{n}$ ( $1 ⩽ n ⩽ 3$ ). We prove the existence and uniqueness of the solution, and also the existence of the global attractor. this work is structured as follows.

We firstly have setting of the problem followed by a priori estimates which allows us to show the existence and uniqueness of the solution of problem, and to assert the dissipativity of the semigroup associated to the problem. And we finally prove the existence of the global attractor, to finish with a conclusion.

2. Setting of the problem

We consider the following Hyperbolic pertubed equation of Cahn–Hilliard with a proliferation term $\begin{array}{l} (2.1) & ε \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} + Δ^{2} u - Δ f (u) + g (u) = 0 in R_{+}^{*} \times Ω, \\ (2.2) & u = Δ u = 0 on R_{+}^{*} \times \partial Ω, \\ (2.3) & u (0, x) = u_{0} (x); \frac{\partial u (0, x)}{\partial t} = u_{1} (x), \forall x \in Ω, \end{array}$ in a bounded and regular domain $Ω \subset R^{n}$ ( $n = 1, 2$ or 3) with boundary $\partial Ω$ .

Here $ε > 0$ is a relaxation parameter, $u = u (t, x)$ the order parameter, f is a regular potential and g is a proliferation term.

Here, f and g satisfies the following assumptions: $\begin{array}{l} (2.4) & f \in C^{2} (R), f (0) = 0, \\ (2.5) & f^{'} (s) ⩾ - c_{0}, s \in R, c_{0} ⩾ 0, \\ (2.6) & f (s) s ⩾ c_{1} F (s) - c_{2} ⩾ - c_{3}, s \in R, c_{1} > 0, c_{2}, c_{3} ⩾ 0, \\ (2.7) & - κ ⩽ \frac{a_{2 p - 1}}{2 p} s^{2 p - 2} - c_{1} ⩽ f^{'} (s) ⩽ 3 p a_{2 p - 1} s^{2 p - 2} + c_{1}, s \in R and κ > 0, c_{1} > 0 . \end{array}$

Remark 2.1.
In this article, we assume $f (s) = \sum_{k = 1}^{2 p - 1} a_{k} s^{k}$ , $a_{2 p - 1} > 0$ , $p ⩾ 2$ and $g (s) = \sum_{i = 1}^{q} b_{i} s^{i}$ , $b_{q} > 0$ , $q ⩾ 1$ . We suppose that $q ⩽ p$ .
Notations.
We denote by $(\cdot, \cdot)$ the usual $L^{2}$ -scalar product with associated norm $‖ \cdot ‖$ , and we set $‖ \cdot ‖_{- k} = ‖ {(- Δ)}^{- \frac{k}{2}} \cdot ‖$ , where $k > 0$ and $- Δ$ denotes the minus Laplace operator with Dirichlet boundary conditions. More generally, $‖ \cdot ‖_{X}$ denotes the norm in the Banach space X.

The same letter c, $c^{'}$ and $c^{″}$ denote constants which may vary from line to line, or even in a same line. Similary, the same letter Q denotes monotone increasing functions which may vary from line to line, or even in a same line.
3. A priori estimates

We multiply (2.1) by ${(- Δ)}^{- 1} \frac{\partial u}{\partial t}$ and integrate over Ω. We obtain $\begin{matrix} \frac{ε}{2} \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + (f (u), \frac{\partial u}{\partial t}) + (g (u), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) = 0, \end{matrix}$ which yields $\begin{matrix} (3.1) & \frac{1}{2} \frac{d}{d t} (ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} ⩽ | (g (u), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) | . \end{matrix}$ We note that $\begin{array}{rcl} | (g (u), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) | & ⩽ & ‖ (g (u), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) ‖ \\ (3.2) & ⩽ & {(\int_{Ω} {(1 + | u |^{q})}^{\frac{6}{5}} d x)}^{\frac{5}{6}} {‖ {(- Δ)}^{- 1} \frac{\partial u}{\partial t} ‖}_{L^{6}} . \end{array}$ Besides $H^{1} (Ω) \subset L^{6} (Ω)$ with continuous injection, so that $\begin{array}{rcl} | (g (u), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) | & ⩽ & c {(\int_{Ω} {(1 + | u |^{q})}^{\frac{6}{5}} d x)}^{\frac{5}{6}} {‖ \frac{\partial u}{\partial t} ‖}_{- 1} \\ ⩽ & c {(\int_{Ω} (1 + | u |^{\frac{6 q}{5}}) d x)}^{\frac{10}{6}} + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} \\ ⩽ & c {({({(\int_{Ω} (1 + | u |^{\frac{6 q}{5}}) d x)}^{\frac{10}{6}})}^{\frac{6}{10}} {(\int_{Ω} d x)}^{\frac{4}{10}})}^{\frac{10}{6}} + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} \\ ⩽ & c {({(\int_{Ω} (1 + | u |^{2 q}) d x)}^{\frac{6}{10}})}^{\frac{10}{6}} + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} \\ (3.3) & ⩽ & c \int_{Ω} (1 + | u |^{2 q}) d x + \frac{1}{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} . \end{array}$ Inserting (3.3) into (3.1), we obtain $\begin{matrix} (3.4) & \frac{d}{d t} (ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} ⩽ c \int_{Ω} (1 + | u |^{2 q}) d x . \end{matrix}$ Owing to Young inequality, we have $\begin{matrix} (3.5) & \int_{Ω} | u |^{2 q} d x ⩽ δ \int_{Ω} F (u) d x + c_{δ}, \forall δ > 0 . \end{matrix}$ Thus $\begin{matrix} (3.6) & \frac{d}{d t} (ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} ⩽ δ_{1} \int_{Ω} F (u) d x + c, δ_{1} > 0 . \end{matrix}$ We multiply (2.1) by ${(- Δ)}^{- 1} u$ and integrate over Ω, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ u ‖_{- 1}^{2} + 2 ε ({(- Δ)}^{\frac{- 1}{2}} \frac{\partial u}{\partial t}, {(- Δ)}^{\frac{- 1}{2}} u)) + ‖ \nabla u ‖^{2} + (f (u), u) \\ (3.7) & ⩽ ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + | (g (u), {(- Δ)}^{- 1} u) | . \end{array}$ Proceeding as in (3.3), we have $\begin{matrix} (3.8) & | (g (u), {(- Δ)}^{- 1} u) | ⩽ c \int_{Ω} (1 + | u |^{2 q}) d x + \frac{1}{2} ‖ \nabla u ‖^{2} . \end{matrix}$ Choosing in (3.5) δ such that $c δ < \frac{c_{1}}{2}$ , (3.7) yields $\begin{matrix} (3.9) & \frac{d}{d t} (‖ u ‖_{- 1} + 2 ε ({(- Δ)}^{\frac{- 1}{2}} \frac{\partial u}{\partial t}, {(- Δ)}^{\frac{- 1}{2}} u)) + ‖ \nabla u ‖^{2} + c_{1} \int_{Ω} F (u) d x ⩽ 2 ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + c^{'} . \end{matrix}$ Summing (3.6) and $λ_{1}$ (3.9), whère $λ_{1} > 0$ , such that $λ_{1} - 2 > 0$ we have $\begin{matrix} (3.10) & \frac{d}{d t} E (t) + c E (t) ⩽ c^{'}, \end{matrix}$ where $\begin{matrix} E (t) = ‖ \nabla u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \int_{Ω} F (u) d x + λ_{1} ‖ u ‖_{- 1}^{2} + 2 ε λ_{1} ({(- Δ)}^{\frac{- 1}{2}} \frac{\partial u}{\partial t}, {(- Δ)}^{\frac{- 1}{2}} u) \end{matrix}$ satisfies $\begin{matrix} (3.11) & E (t) ⩾ c (‖ \nabla u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \int_{Ω} F (u) d x) + c^{'}, c > 0 \end{matrix}$ and $\begin{matrix} (3.12) & E (t) ⩽ c^{″} (‖ \nabla u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + \int_{Ω} F (u) d x) + c^{‴}, c^{″} > 0 . \end{matrix}$ We find from (3.10) and Gronwall’s lemma the dissipative estimate $\begin{matrix} (3.13) & E (t) ⩽ c e^{- c t} (‖ u_{0} ‖_{H^{1}}^{2} + ε {‖ \frac{\partial u_{0}}{\partial t} ‖}_{- 1}^{2} + \int_{Ω} F (u_{0}) d x) + c, c > 0, t ⩾ 0 . \end{matrix}$ We multiply (2.1) by $\frac{\partial u}{\partial t}$ and integrate over Ω, we obtain $\begin{matrix} \frac{ε}{2} \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}^{2} + \frac{1}{2} \frac{d}{d t} ‖ Δ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} = (Δ f (u), \frac{\partial u}{\partial t}) - (g (u), \frac{\partial u}{\partial t}), \end{matrix}$ which yields $\begin{matrix} (3.14) & \frac{d}{d t} (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) + {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ c ({‖ Δ f (u) ‖}^{2} + {‖ g (u) ‖}^{2}) . \end{matrix}$ Let us note that $H^{2} (Ω) \subset L^{\infty} (Ω)$ and $H^{2} (Ω)$ is continuously embedded in $C (\overline{Ω})$ , then $\begin{array}{rcl} (3.15) & {‖ Δ f (u) ‖}^{2} + {‖ g (u) ‖}^{2} ⩽ Q (‖ u ‖_{H^{2}}) . \end{array}$ Substituting (3.15) in (3.14), we have $\begin{matrix} (3.16) & \frac{d}{d t} (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) + {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ Q (‖ u ‖_{H^{2}}), \end{matrix}$ in particular $\begin{matrix} (3.17) & \frac{d}{d t} (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) ⩽ Q (‖ Δ u ‖^{2}) . \end{matrix}$ We set $\begin{matrix} y = ‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}, \end{matrix}$ we deduce from (3.17) an inequation of the form $\begin{matrix} (3.18) & y^{'} ⩽ Q (y), y (0) = ‖ Δ u_{0} ‖^{2} + ε ‖ u_{1} ‖^{2} . \end{matrix}$ Let z be the solution to the ordinary differential equation $\begin{matrix} z^{'} = Q (z), z (0) = y (0) = ‖ Δ u_{0} ‖^{2} + ε ‖ u_{1} ‖^{2} . \end{matrix}$ It follows from the comparison principle, that there exists a time $T_{0} = T_{0} (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) > 0$ belonging to, say $(0, \frac{1}{2})$ such that $\begin{matrix} y (t) ⩽ z (t) \forall t ⩽ T_{0} \end{matrix}$ hence $\begin{matrix} (3.19) & {‖ u (t) ‖}_{H^{2}}^{2} + ε {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \forall t ⩽ T_{0} . \end{matrix}$ We multiply (2.1) by ${(- Δ)}^{- 2} \frac{\partial^{2} u}{\partial t^{2}}$ and integrate over Ω, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}_{- 2}^{2} + ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} + (u, \frac{\partial^{2} u}{\partial t^{2}}) \\ = - (f (u), {(- Δ)}^{- 1} \frac{\partial^{2} u}{\partial t^{2}}) - ({(- Δ)}^{- 1} g (u), {(- Δ)}^{- 1} \frac{\partial^{2} u}{\partial t^{2}}), \end{array}$ which gives, owing to (3.15) and (3.19) $\begin{matrix} (3.20) & \frac{1}{2} \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}_{- 2}^{2} + ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} + (u, \frac{\partial^{2} u}{\partial t^{2}}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + \frac{ε}{2} {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} . \end{matrix}$ Thus $\begin{matrix} (3.21) & \frac{d}{d t} ({‖ \frac{\partial u}{\partial t} ‖}_{- 2}^{2} + 2 (u, \frac{\partial u}{\partial t})) + ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + 2 {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{matrix}$ Summing (3.16) and $λ_{2}$ (3.21), where $λ_{2} > 0$ and $1 - 2 λ_{2} > 0$ , we have $\begin{matrix} (3.22) & \frac{d}{d t} E_{1} (t) + c ({‖ \frac{\partial u}{\partial t} ‖}^{2} + ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), c > 0, \end{matrix}$ where $\begin{matrix} E_{1} (t) = ‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} + λ_{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 2}^{2} + 2 λ_{2} (u, \frac{\partial u}{\partial t}) . \end{matrix}$ We have $\begin{matrix} (3.23) & E_{1} (t) = ‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} + λ_{2} {‖ \frac{\partial u}{\partial t} ‖}_{- 2}^{2} + 2 λ_{2} (u, \frac{\partial u}{\partial t}) ⩾ c (‖ u ‖_{H^{2}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) . \end{matrix}$ Integrating (3.22) from 0 to t and owing to (3.23), we obtain, considering $t ⩽ T_{0}$ $\begin{matrix} (3.24) & c (‖ u ‖_{H^{2}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) + c \int_{0}^{t} ({‖ \frac{\partial u (s)}{\partial t} ‖}^{2} + ε {‖ \frac{\partial^{2} u (s)}{\partial t^{2}} ‖}_{- 2}^{2}) d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ we deduce that $\begin{matrix} (3.25) & \int_{0}^{t} ε {‖ \frac{\partial^{2} u (s)}{\partial t^{2}} ‖}_{- 2}^{2} d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) \end{matrix}$ and $\begin{matrix} (3.26) & \int_{0}^{t} {‖ \frac{\partial u (s)}{\partial t} ‖}^{2} d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ We now differentiate (2.1) with respect to time and rewrite the resulting equation as $\begin{matrix} (3.27) & ε {(- Δ)}^{- 1} \frac{\partial^{2} θ}{\partial t^{2}} + {(- Δ)}^{- 1} \frac{\partial θ}{\partial t} - Δ θ + f^{'} (u) θ + {(- Δ)}^{- 1} (g^{'} (u) θ) = 0, \end{matrix}$ where $θ = \frac{\partial u}{\partial t}$ .

We multiply (3.27) by $t {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}$ and integrate over Ω, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (t (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2})) + t {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} \\ (3.28) & = - t (f^{'} (u) θ, {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) - t ({(- Δ)}^{- 1} (g^{'} (u) θ), {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) + ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2} . \end{array}$ Besides $u \in H^{2} (Ω)$ , then, oving (2.7), we have $\begin{array}{rcl} (f^{'} (u) θ, {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) & ⩽ & \int_{Ω} | f^{'} (u) | | θ | | {(- Δ)}^{- 1} \frac{\partial θ}{\partial t} | d x \\ ⩽ & \int_{Ω} (3 p a_{2 p - 1} | u |^{2 p - 2} + c_{1}) | θ | | {(- Δ)}^{- 1} \frac{\partial θ}{\partial t} | d x \\ ⩽ & (3 p a_{2 p - 1} ‖ u ‖_{L^{\infty}}^{2 p - 2} + c_{1}) ‖ θ ‖ {‖ \frac{\partial θ}{\partial t} ‖}_{- 2} \\ (3.29) & ⩽ & Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) ‖ θ ‖^{2} + \frac{ε}{4} {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} . \end{array}$ We note that, since $H^{2} (Ω) \subset L^{\infty} (Ω)$ with continuous injection, $\begin{array}{rcl} ({(- Δ)}^{- 1} (g^{'} (u) θ), {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) & ⩽ & \int_{Ω} | g^{'} (u) | | θ | | {(- Δ)}^{- 2} \frac{\partial θ}{\partial t} | d x \\ ⩽ & c {‖ g^{'} (u) ‖}_{L^{\infty}} ‖ θ ‖ {‖ \frac{\partial θ}{\partial t} ‖}_{- 2} \\ ⩽ & Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) ‖ θ ‖ {‖ \frac{\partial θ}{\partial t} ‖}_{- 2} \\ (3.30) & ⩽ & Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) ‖ θ ‖^{2} + \frac{ε}{4} {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} . \end{array}$ Inserting (3.29) and (3.30) into (3.28), we find $\begin{matrix} (3.31) & \frac{d}{d t} E_{2} (t) + t {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} ⩽ c E_{2} (t) + ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}, \end{matrix}$ where $\begin{matrix} E_{2} (t) = t (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}) . \end{matrix}$ Applying Gronwall’s lemma to (3.31) and owing to (3.25) and (3.26), we obtain $\begin{matrix} (3.32) & t (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}) + \int_{0}^{t} s e^{c (t - s)} {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Thus $\forall t \in] 0, T_{0}]$ $\begin{matrix} (3.33) & ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} ⩽ \frac{1}{t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) \end{matrix}$ and $\begin{matrix} (3.34) & {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ \frac{1}{t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Multiplying (3.27) by ${(- Δ)}^{- 1} \frac{\partial θ}{\partial t}$ and integrate over Ω, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}) + {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} \\ (3.35) & = - ({(- Δ)}^{- 1} (g^{'} (u) θ), {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) - (f^{'} (u) θ, {(- Δ)}^{- 1} \frac{\partial θ}{\partial t}) . \end{array}$ Thanks to (3.29) and (3.30), we have $\begin{matrix} (3.36) & \frac{1}{2} \frac{d}{d t} (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}) + {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) (ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2}), \end{matrix}$ and using the Gronwall’s lemma, Owing to (3.33) and (3.34), we find $\begin{matrix} (3.37) & ε {‖ \frac{\partial θ}{\partial t} ‖}_{- 2}^{2} + ‖ θ ‖^{2} + \int_{T_{0}}^{t} e^{- c (t - s)} {‖ \frac{\partial θ (s)}{\partial t} ‖}_{- 2}^{2} d s ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ which yields, for all $t ⩾ T_{0}$ $\begin{matrix} (3.38) & ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{- 2}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ and $\begin{matrix} (3.39) & {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ We now rewrite, for $t ⩾ T_{0}$ fixed, (2.1) in the form $\begin{matrix} (3.40) & - Δ u + f (u) + {(- Δ)}^{- 1} g (u) = h_{u} (t), u = 0 on \partial Ω, \end{matrix}$ where $\begin{matrix} (3.41) & h_{u} (t) = - ε {(- Δ)}^{- 1} \frac{\partial^{2} u}{\partial t^{2}} - {(- Δ)}^{- 1} \frac{\partial u}{\partial t} . \end{matrix}$ Owing to (3.38) and (3.39), we obtain $\begin{matrix} (3.42) & {‖ h_{u} (t) ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ We multiply (3.40) by $- Δ u$ and integrate over Ω, we obtain $\begin{matrix} (3.43) & ‖ Δ u ‖^{2} ⩽ c {‖ h_{u} (t) ‖}^{2} + c^{'} ({‖ g (u) ‖}^{2} + {‖ f (u) ‖}^{2}) . \end{matrix}$ Inserting (3.15) into (3.19) and owing to (3.42), we have $\begin{matrix} (3.44) & ‖ Δ u ‖^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Summing ε (3.39) and (3.44), we obtain $\begin{matrix} ‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ then $\begin{matrix} (3.45) & ‖ u ‖_{H^{2}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), c ⩾ 0, t ⩾ T_{0} . \end{matrix}$ Finaly, we deduce from (3.19) and (3.45) that $\begin{matrix} (3.46) & ‖ u ‖_{H^{2}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ e^{c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), c ⩾ 0, t ⩾ 0 . \end{matrix}$ We multiply (2.1) by u and integrate over Ω, we find $\begin{matrix} (3.47) & \frac{d}{d t} (‖ u ‖^{2} + 2 ε (\frac{\partial u}{\partial t}, u)) + ‖ Δ u ‖^{2} ⩽ c ({‖ Δ f (u) ‖}^{2} + {‖ g (u) ‖}^{2}) + 2 ε {‖ \frac{\partial u}{\partial t} ‖}^{2}, \end{matrix}$ which implies, thanks to (3.15) and (3.19) $\begin{matrix} (3.48) & \frac{d}{d t} (‖ u ‖^{2} + 2 ε (\frac{\partial u}{\partial t}, u)) + ‖ Δ u ‖^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + 2 ε {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{matrix}$ Summing (3.16) and $λ_{3}$ (3.48) where $λ_{3} > 0$ such that $1 - 2 λ_{3} > 0$ , we get $\begin{matrix} (3.49) & \frac{d}{d t} E_{4} (t) + c (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'}, c > 0 \end{matrix}$ with $\begin{matrix} E_{4} (t) = ‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2} + λ_{3} ‖ u ‖^{2} + 2 λ_{3} (\frac{\partial u}{\partial t}, u) . \end{matrix}$ There also exists $c, c^{'} > 0$ such that $\begin{matrix} (3.50) & c (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) ⩽ E_{4} (t) ⩽ c^{″} (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) . \end{matrix}$ We deduce, owing to (3.49) and (3.50), that $\begin{matrix} (3.51) & \frac{d}{d t} E_{4} (t) + c (‖ Δ u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'}, c > 0, \end{matrix}$ and it follows from (3.50) and (3.51) that $\begin{matrix} (3.52) & \int_{0}^{1} (‖ u ‖_{H^{2}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}^{2}) d t ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'} . \end{matrix}$ Therefore, the estimate (3.52) allows to affirm that there exists $T \in (0, 1)$ such that $\begin{matrix} (3.53) & {‖ u (T) ‖}_{H^{2}}^{2} + ε {‖ \frac{\partial u (T)}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'} . \end{matrix}$ Actually, repeating the above estimates, starting from $t = T$ instead of $t = 0$ , we see that (3.53) holds for $T = 1$ , i.e. we have the smoothing property $\begin{matrix} (3.54) & {‖ u (1) ‖}_{H^{2}}^{2} + ε {‖ \frac{\partial u (1)}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'} . \end{matrix}$ According to the smoothing property, it is not difficult to prove that we have, owing to (3.13), (3.46) and (3.54), the dissipative following estimate $\begin{matrix} (3.55) & {‖ u (t) ‖}_{H^{2}}^{2} + ε {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'}, c > 0, t ⩾ 0 . \end{matrix}$

4. Existence and uniqueness of solutions

We first have the following theorem

Theorem 4.1 (Existence).

We assume that $(u_{0}, u_{1}) \in (H^{2} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω)) \times L^{2} (Ω)$ . Then, the problem ( 2.1 )–( 2.3 ) possesses at least one solution u such that $u \in L^{\infty} (0, T; H^{2} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω))$ , $\frac{\partial u}{\partial t} \in L^{\infty} (0, T; L^{2} (Ω))$ and $\frac{\partial^{2} u}{\partial t^{2}} \in L^{2} (0, T; H^{- 2} (Ω))$ , $\forall T > 0$ .

The proof of existence is based on the priori estimates and a standard Galerkin scheme.

Theorem 4.2 (Uniqueness).

We assume that $(u_{0}, u_{1}) \in (H^{2} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω)) \times L^{2} (Ω)$ . Then, the system ( 2.1 )–( 2.3 ) possesses a unique solution u such that $u \in L^{\infty} (0, T; H^{2} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω))$ , $\frac{\partial u}{\partial t} \in L^{\infty} (0, T; L^{2} (Ω))$ and $\frac{\partial^{2} u}{\partial t^{2}} \in L^{2} (0, T; H^{- 2} (Ω))$ , $\forall T > 0$ .

Proof.
Let now $u^{(1)}$ and $u^{(2)}$ be two solutions of (2.1)–(2.3), with initial datas $(u_{0}^{(1)}, u_{1}^{(1)})$ and $(u_{0}^{(2)}, u_{1}^{(2)}) \in (H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times L^{2} (Ω)$ . We set $u = u^{(1)} - u^{(2)}$ , $u_{0} = u_{0}^{(1)} - u_{0}^{(2)}$ and $u_{1} = u_{1}^{(1)} - u_{1}^{(2)}$ , then u verifies the following problem $\begin{array}{l} (4.1) & ε \frac{\partial^{2} u}{\partial t} + \frac{\partial u}{\partial t} + Δ^{2} u - Δ (f (u^{(1)}) - f (u^{(2)})) + g (u^{(1)}) - g (u^{(2)}) = 0 in [0, T] \times Ω, \\ (4.2) & Δ u = u = 0 on [0, T] \times \partial Ω, \\ (4.3) & u (0, x) = u_{0} (x); \frac{\partial u (0, x)}{\partial t} = u_{1} (x), \forall x \in Ω . \end{array}$ We multiply (4.1) by ${(- Δ)}^{- 1} \frac{\partial u}{\partial t}$ and integrate over Ω. We obtain $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + ‖ \nabla u ‖^{2}) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} & = & - (g (u^{(1)}) - g (u^{(2)}), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) \\ (4.4) & - (\nabla (f (u^{(1)}) - f (u^{(2)})), {(- Δ)}^{- \frac{1}{2}} \frac{\partial u}{\partial t}) . \end{array}$ We again note that $\begin{matrix} (4.5) & (\nabla (f (u^{(1)}) - f (u^{(2)})), {(- Δ)}^{- \frac{1}{2}} \frac{\partial u}{\partial t}) ⩽ ‖ \nabla (f (u^{(1)}) - f (u^{(2)})) ‖ ‖ {(- Δ)}^{- \frac{1}{2}} \frac{\partial u}{\partial t} ‖ . \end{matrix}$ Besides $\begin{array}{rcl} f (u^{(1)}) - f (u^{(2)}) & = & \sum_{k = 1}^{2 p - 1} a_{k} ({(u^{(1)})}^{k} - {(u^{(2)})}^{k}) \\ = & (u^{(1)} - u^{(2)}) (a_{1} + a_{2} (u^{(1)} + u^{(2)}) + \sum_{k = 3}^{2 p - 1} a_{k} \sum_{j = 0}^{k - 1} {(u^{(1)})}^{k - j - 1} \cdot {(u^{(2)})}^{j}) \end{array}$ which gives $\begin{array}{rcl} \nabla (f (u^{(1)}) - f (u^{(2)})) & = & \nabla u (a_{1} + a_{2} (u^{(1)} + u^{(2)}) + \sum_{k = 3}^{2 p - 1} a_{k} \sum_{j = 0}^{k - 1} {(u^{(1)})}^{k - j - 1} {(u^{(2)})}^{j}) \\ + u (a_{1} + a_{2} (\nabla u^{(1)} + \nabla u^{(2)}) \\ + \sum_{k = 3}^{2 p - 2} a_{k} \sum_{j = 0}^{k - 2} (k - j - 1) \nabla u^{(1)} {(u^{(1)})}^{k - j - 2} {(u^{(2)})}^{j} \\ + \sum_{k = 3}^{2 p - 2} a_{k} \sum_{j = 1}^{k - 1} j \nabla u^{(2)} {(u^{(1)})}^{k - j - 1} \cdot {(u^{(2)})}^{j - 1}) \end{array}$ which implies $\begin{array}{l} | (\nabla f (u^{(1)}) - f (u^{(2)})) | \\ ⩽ | \nabla u | (| a_{1} | + | a_{2} | (| u^{(1)} | + | u^{(2)} |) + \sum_{k = 3}^{2 p - 1} | a_{k} | \sum_{j = 0}^{k - 1} ({| u^{(1)} |}^{k - j - 1} {| u^{(2)} |}^{j})) \\ + | u | (| a_{2} | (| \nabla u^{(1)} | + | \nabla u^{(2)} |) + | \nabla u^{(1)} | \sum_{k = 3}^{2 p - 2} | a_{k} | \sum_{j = 0}^{k - 2} (k - j - 1) {| u^{(1)} |}^{k - j - 2} {| u^{(2)} |}^{j} \\ + | \nabla u^{(2)} | \sum_{k = 3}^{2 p - 2} | a_{k} | \sum_{j = 1}^{k - 1} j {| u^{(1)} |}^{k - j - 1} {| u^{(2)} |}^{j - 1}) . \end{array}$ There exists $C > 0$ such that $| a_{k} | ⩽ C$ , $\forall k \in {1, 2, \dots, 2 p - 1}$ .

Thus, we find $\begin{array}{l} | \nabla (f (u^{(1)}) - f (u^{(2)})) | \\ ⩽ | \nabla u | (| a_{1} | + | a_{2} | (| u^{(1)} | + | u^{(2)} |) + \sum_{k = 3}^{2 p - 1} | a_{k} | \frac{k}{2} ({| u^{(1)} |}^{k - 1} + {| u^{(2)} |}^{k - 1})) \\ + | u | (| \nabla u^{(1)} | + | \nabla u^{(2)} |) (| a_{2} | + \sum_{k = 3}^{2 p - 2} | a_{k} | \frac{k (k - 1)}{2} ({| u^{(1)} |}^{k - 2} + {| u^{(2)} |}^{k - 2})) \end{array}$ which implies $\begin{array}{l} | \nabla f (u^{(1)}) - f (u^{(2)}) | \\ ⩽ C | \nabla u | (| a_{1} | + | a_{2} | (| u^{(1)} | + | u^{(2)} |) + \sum_{k = 3}^{2 p - 1} ({| u^{(1)} |}^{k - 1} + {| u^{(2)} |}^{k - 1})) \\ + C | u | (| \nabla u^{(1)} | + | \nabla u^{(2)} |) (| a_{2} | + \sum_{k = 3}^{2 p - 2} ({| u^{(1)} |}^{k - 2} + {| u^{(2)} |}^{k - 2})) . \end{array}$ Noting that $u^{(1)}, u^{(2)} \in H^{2} (Ω)$ , then $u^{(1)}, u^{(2)} \in C (\overline{Ω})$ , there exists $C_{1}, C_{2} > 0$ such that ${sup}_{(t, x) \in (0, T) \times Ω} | u^{(1)} (t, x) | ⩽ C_{1}$ and ${sup}_{(t, x) \in (0, T) \times Ω} | u^{(2)} (t, x) | ⩽ C_{2}$ , then we have $\begin{array}{l} ‖ \nabla (f (u^{(1)}) - f (u^{(2)})) ‖ \\ ⩽ C (| a_{1} | + | a_{2} | ({‖ u^{(1)} ‖}_{L^{\infty}} + {‖ u^{(2)} ‖}_{L^{\infty}}) + \sum_{k = 3}^{2 p - 1} ({‖ u^{(1)} ‖}_{L^{\infty}}^{k - 1} + {‖ u^{(2)} ‖}_{L^{\infty}}^{k - 1})) ‖ \nabla u ‖ \\ (4.6) & + C^{'} ‖ u ‖_{L^{4}} ({‖ \nabla u^{(1)} ‖}_{L^{4}} + {‖ \nabla u^{(2)} ‖}_{L^{4}}) (| a_{2} | + \sum_{k = 3}^{2 p - 2} ({‖ u^{(1)} ‖}_{L^{\infty}}^{k - 2} + {‖ u^{(2)} ‖}_{L^{\infty}}^{k - 2})) . \end{array}$ Remark that $H^{1} (Ω) \subset L^{4} (Ω)$ with continuous injection and thanks to (3.19), we find $\begin{matrix} (4.7) & ‖ \nabla (f (u^{(1)}) - f (u^{(2)})) ‖ ⩽ Q ({‖ u_{0}^{(1)} ‖}_{H^{2}}, {‖ u_{0}^{(2)} ‖}_{H^{2}}, ‖ u_{1}^{(1)} ‖, ‖ u_{1}^{(2)} ‖) ‖ \nabla u ‖ . \end{matrix}$ Hence $\begin{array}{l} \int_{Ω} | \nabla (f (u^{(1)}) - f (u^{(2)})) | | {(- Δ)}^{- 1} \frac{\partial u}{\partial t} | d x \\ (4.8) & ⩽ Q ({‖ u_{0}^{(1)} ‖}_{H^{2}}, {‖ u_{0}^{(2)} ‖}_{H^{2}}, ‖ u_{1}^{(1)} ‖, ‖ u_{1}^{(2)} ‖) ‖ \nabla u ‖^{2} + \frac{1}{4} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} . \end{array}$ We remark that $\begin{matrix} (4.9) & | (g (u^{(1)}) - g (u^{(2)}), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) | ⩽ ‖ g (u^{(1)}) - g (u^{(2)}) ‖ | {(- Δ)}^{- 1} \frac{\partial u}{\partial t} ‖ . \end{matrix}$ Besides $\begin{array}{rcl} g (u^{(1)}) - g (u^{(2)}) & = & \sum_{i = 1}^{q} b_{i} ({(u^{(1)})}^{i} - {(u^{(2)})}^{i}) \\ = & (u^{(1)} - u^{(2)}) (b_{1} + b_{2} (u^{(1)} + u^{(2)}) + \sum_{i = 3}^{q - 1} b_{i} \sum_{j = 0}^{i - 1} {(u^{(1)})}^{i - j - 1} \cdot {(u^{(2)})}^{j}) \end{array}$ which implies $\begin{matrix} | g (u^{(1)}) - g (u^{(2)}) | ⩽ | u | (| b_{1} | + | b_{2} | (| u^{(1)} | + | u^{(2)} |) + \sum_{i = 3}^{q - 1} | b_{i} | \sum_{j = 0}^{i - 1} {| u^{(1)} |}^{i - j - 1} \cdot {| u^{(2)} |}^{j}) . \end{matrix}$ There exists $C > 0$ such that $| b_{i} | ⩽ C$ , $\forall i \in {1, 2, \dots, q}$ .

Applying Young’s inequality, we obtain $\begin{array}{rcl} {| u^{(1)} |}^{i - j - 1} \cdot {| u^{(2)} |}^{j} & ⩽ & \frac{i - 1 - j}{i - 1} {({| u^{(1)} |}^{i - j - 1})}^{\frac{k - 1}{i - j - 1}} + \frac{j}{i - 1} {({| u^{(2)} |}^{j})}^{\frac{i - 1}{j}} \\ ⩽ & \frac{i - 1 - j}{i - 1} {| u^{(1)} |}^{i - 1} + \frac{j}{i - 1} {| u^{(2)} |}^{i - 1}, \end{array}$ which yields $\begin{array}{rcl} \sum_{j = 0}^{i - 1} {| u^{(1)} |}^{i - j - 1} \cdot {| u^{(2)} |}^{j} & ⩽ & \sum_{j = 0}^{i - 1} \frac{i - 1 - j}{i - 1} {| u^{(1)} |}^{i - 1} + \sum_{j = 0}^{i - 1} \frac{j}{i - 1} {| u^{(2)} |}^{i - 1} \\ ⩽ & \frac{i}{2} ({| u^{(1)} |}^{i - 1} + {| u^{(2)} |}^{i - 1}) . \end{array}$ Then we get $\begin{matrix} | g (u^{(1)}) - g (u^{(2)}) | ⩽ | u | (| b_{1} | + | b_{2} | (| u^{(1)} | + | u^{(2)} |) + \sum_{i = 3}^{q - 1} \frac{i}{2} | b_{i} | ({| u^{(1)} |}^{i - 1} + {| u^{(2)} |}^{i - 1})) \end{matrix}$ which implies $\begin{array}{rcl} ‖ g (u^{(1)}) - g (u^{(2)}) ‖ & ⩽ & ‖ u ‖ (| b_{1} | + | b_{2} | ({‖ u^{(1)} ‖}_{L^{\infty}} + {‖ u^{(2)} ‖}_{L^{\infty}}) \\ + \sum_{i = 3}^{q - 1} \frac{i}{2} | b_{i} | ({‖ u^{(1)} ‖}_{L^{\infty}}^{i - 1} + {‖ u^{(2)} ‖}_{L^{\infty}}^{i - 1})) . \end{array}$ Noting that $H^{2} (Ω) \subset L^{\infty} (Ω)$ with continuous injection and while using (3.46) and poincaré inequality, we have $\begin{array}{rcl} | (g (u^{(1)}) - g (u^{(2)}), {(- Δ)}^{- 1} \frac{\partial u}{\partial t}) | & ⩽ & Q ({‖ u_{0}^{1} ‖}_{H^{2}}, {‖ u_{0}^{2} ‖}_{H^{2}}, ‖ u_{1}^{1} ‖, ‖ u_{1}^{2} ‖) ‖ \nabla u ‖ {‖ \frac{\partial u}{\partial t} ‖}_{- 1} \\ (4.10) & ⩽ & Q ({‖ u_{0}^{1} ‖}_{H^{2}}, {‖ u_{0}^{2} ‖}_{H^{2}}, ‖ u_{1}^{1} ‖, ‖ u_{1}^{2} ‖) ‖ \nabla u ‖^{2} + \frac{1}{4} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} . \end{array}$ Inserting (4.8) and (4.10) into (4.4), we find $\begin{array}{l} \frac{d}{d t} (‖ \nabla u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2}) + {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} \\ (4.11) & ⩽ Q ({‖ u_{0}^{(1)} ‖}_{H^{2}}, {‖ u_{0}^{(2)} ‖}_{H^{2}}, ‖ u_{1}^{(1)} ‖, ‖ u_{1}^{(2)} ‖) (‖ \nabla u ‖^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2}) . \end{array}$ Where Q is monotone increasing with respect to both arguments. We deduce from (4.11) and Gronwall’s lemma that $\begin{matrix} (4.12) & {‖ \nabla u (t) ‖}^{2} + ε {‖ \frac{\partial u (t)}{\partial t} ‖}_{- 1}^{2} ⩽ e^{c t} (‖ u_{0} ‖_{H^{1}}^{2} + ε ‖ u_{1} ‖_{- 1}^{2}), \forall t \in [0, T], \end{matrix}$ hence the uniqueness, as well as the continuous depending with respect to the initial data. □
Theorem 4.3 (Regularity).

We assume $(u_{0}, u_{1}) \in (H^{3} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω)$ , $ε < 1$ . Then the problem ( 2.1 )–( 2.3 ) possesses a unique solution u such that $\forall T > 0$ $\begin{matrix} u \in L^{\infty} (0, T; H^{3} (Ω) \cap H_{0}^{1} (Ω)), \frac{\partial u}{\partial t} \in L^{\infty} (0, T; H_{0}^{1} (Ω)) and \frac{\partial^{2} u}{\partial t^{2}} \in L^{2} (0, T; H^{- 1} (Ω)) . \end{matrix}$

Proof.
We multiply (2.1) by $- Δ \frac{\partial u}{\partial t}$ and intregrate over Ω, we obtain $\begin{matrix} (4.13) & \frac{1}{2} \frac{d}{d t} (ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla Δ u ‖^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} = - (Δ f (u), Δ \frac{\partial u}{\partial t}) + (g (u), Δ \frac{\partial u}{\partial t}) . \end{matrix}$ Note that $H^{2} (Ω) \subset L^{\infty} (Ω)$ , $H^{2} (Ω) \subset C (\overline{Ω})$ and $H^{1} (Ω) \subset L^{6} (Ω)$ with continuous injection, and owing to the fact that $f \in C^{2} (R)$ , $g \in C^{1} (R)$ and grace (3.16), then $\begin{array}{l} (Δ f (u), Δ \frac{\partial u}{\partial t}) \\ ⩽ ‖ \nabla (Δ f (u)) ‖ ‖ \nabla \frac{\partial u}{\partial t} ‖ \\ ⩽ {‖ \nabla (f^{″} (u) {(\nabla u)}^{2} + f^{'} (u) Δ u) ‖}^{2} + \frac{1}{4} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} \\ ⩽ c ({‖ f^{‴} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla u ‖_{L^{6}}^{6} + {‖ f^{″} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla u ‖_{L^{4}}^{2} ‖ Δ u ‖_{L^{4}}^{2} + {‖ f^{'} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla Δ u ‖_{L^{2}}^{2}) + \frac{1}{4} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} \\ (4.14) & ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'} ‖ \nabla Δ u ‖^{2} + \frac{1}{4} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} \end{array}$ and $\begin{array}{rcl} (g (u), Δ \frac{\partial u}{\partial t}) & = & (g^{'} (u) \nabla u, \nabla \frac{\partial u}{\partial t}) \\ ⩽ & \int_{Ω} | g^{'} (u) | | \nabla u | | \nabla \frac{\partial u}{\partial t} | d x \\ ⩽ & {‖ g^{'} (u) ‖}_{L^{\infty}} ‖ \nabla u ‖ ‖ \nabla \frac{\partial u}{\partial t} ‖ \\ ⩽ & Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) ‖ u ‖_{H^{2}} ‖ \nabla \frac{\partial u}{\partial t} ‖ \\ (4.15) & ⩽ & Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + \frac{1}{4} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$ Inserting (4.14) and (4.15) into (4.13), we find $\begin{matrix} (4.16) & \frac{d}{d t} (‖ \nabla Δ u ‖^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}) + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c ‖ \nabla Δ u ‖^{2} . \end{matrix}$ We now multiply (2.1) by $- Δ u$ and integrate over Ω, we have $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖^{2} + ‖ \nabla Δ u ‖^{2} + (ε \frac{\partial^{2} u}{\partial t^{2}}, Δ u) ⩽ c ({‖ Δ f (u) ‖}^{2} + {‖ g (u) ‖}^{2}) + \frac{1}{2} ‖ \nabla Δ u ‖^{2}, \end{matrix}$ which implies $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖^{2} + ‖ Δ \nabla u ‖^{2} + \frac{d}{d t} (ε \frac{\partial u}{\partial t}, Δ u) ⩽ c ({‖ Δ f (u) ‖}^{2} + {‖ g (u) ‖}^{2}) + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} . \end{matrix}$ Thanks to (3.15) and (3.46), we find $\begin{matrix} (4.17) & \frac{d}{d t} (‖ \nabla u ‖^{2} + 2 ε (\frac{\partial u}{\partial t}, Δ u)) + ‖ \nabla Δ u ‖^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + 2 ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} . \end{matrix}$ Summing (4.16) and (4.17), we obtain $\begin{matrix} (4.18) & \frac{d}{d t} E_{3} (t) + c (‖ \nabla Δ u ‖^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c (‖ \nabla Δ u ‖^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}), \end{matrix}$ where $\begin{matrix} E_{3} (t) = ‖ \nabla Δ u ‖^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + (‖ \nabla u ‖^{2} + 2 ε (\frac{\partial u}{\partial t}, Δ u)) . \end{matrix}$ We also have $\begin{matrix} (4.19) & c (‖ u ‖_{H^{3}}^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}) ⩽ E_{3} (t) ⩽ c^{″} (‖ u ‖_{H^{3}}^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}), c, c^{″} ⩾ 0 . \end{matrix}$ Thus, we deduce thanks to (4.18) and (4.19) the following estimate $\begin{matrix} (4.20) & \frac{d}{d t} E_{3} (t) - c E_{3} (t) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Applying the Gronwall’s lemma and owing to (4.19), we have $\begin{array}{rcl} (4.21) & ‖ u ‖_{H^{3}}^{2} + ε {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + (‖ u_{0} ‖_{H^{3}}^{2} + ε ‖ u_{1} ‖_{H^{1}}^{2}) e^{C T} . \end{array}$ Therefore, we deduce that $\forall T > 0$ $\begin{matrix} u \in L^{\infty} (0, T; H^{3} (Ω) \cap H_{0}^{1} (Ω)) and \frac{\partial u}{\partial t} \in L^{\infty} (0, T; H_{0}^{1} (Ω)) . \end{matrix}$ Multiply (2.1) by ${(- Δ)}^{- 1} \frac{\partial^{2} u}{\partial t^{2}}$ and integrate over Ω. We find, owing to (3.15) $\begin{matrix} (4.22) & \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}_{H^{- 1}}^{2} + ε {‖ \frac{\partial^{2} u}{\partial t^{2}} ‖}_{H^{- 1}}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c^{'} ‖ \nabla Δ u ‖^{2} . \end{matrix}$ Intgrating from 0 to t, and owing to (4.21), we find $\begin{matrix} (4.23) & {‖ \frac{\partial u}{\partial t} ‖}_{H^{- 1}}^{2} + ε \int_{0}^{t} {‖ \frac{\partial^{2} u (s)}{\partial t^{2}} ‖}_{H^{- 1}}^{2} d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Finally, $\begin{matrix} \frac{\partial^{2} u}{\partial t^{2}} \in L^{2} (0, T; H^{- 1} (Ω)) . \end{matrix}$ From there the theorem. □

We have, thanks to the Theorems 4.1 and 4.3, two respective phase spaces $\begin{matrix} ϕ_{2} = (H^{2} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω)) \times L^{2} (Ω) \end{matrix}$ and $\begin{matrix} ϕ_{3} = (H^{3} (Ω) \cap H_{0}^{1} (Ω) \cap L^{2 p} (Ω)) \times H_{0}^{1} (Ω) \end{matrix}$ and we have two energy norms for the problem (2.1)–(2.3), in those phase spaces $\begin{matrix} {‖ (u, \frac{\partial u}{\partial t}) ‖}_{ϕ_{k}}^{2} = ‖ u ‖_{H^{k}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{H^{k - 2}}^{2}, k = 2, 3 . \end{matrix}$ We then define the continuous semigroup $\begin{array}{l} S_{ε} (t) : & ϕ_{k} ⟼ ϕ_{k} \\ (u_{0}, u_{1}) ⟼ (u (t), \frac{\partial u (t)}{\partial t}), \end{array}$ for $k = 2, 3$ , where u is the unique solution of problem (2.1)–(2.3) with initial data $(u_{0}, u_{1}) \in ϕ_{k}$ .
Corollary 4.1.
For $ε < 1$ , the semigroup of operators $S_{ε} (t)$ , $t ⩾ 0$ associated to the problem ( 2.1 )–( 2.3 ) is dissipative in $ϕ_{2}$ .
Proof.
Taking $t_{0} = t_{0} (R_{0}) = \frac{- 1}{β} ln (\frac{C}{Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖)})$ we have, thanks to estimate (3.55) and for all $t ⩾ t_{0}$ , the estimate below $\begin{matrix} (4.24) & {‖ (u (t), \frac{\partial u (t)}{\partial t}) ‖}_{ϕ_{2}} ⩽ \sqrt{2 C}, \end{matrix}$ which concludes the proof. □

We denote $\begin{matrix} B_{R_{0}}^{1} (ϵ) = {(u, \frac{\partial u}{\partial t}) \in ϕ_{2} / {‖ (u, \frac{\partial u}{\partial t}) ‖}_{ϕ_{2}} ⩽ R_{0}}, \end{matrix}$ the bounded absorbing set for $S_{ε} (t)$ in phase space $ϕ_{2}$ , where $R_{0}$ is large enough.
5. Dissipativity and regularity of the semigroup

The dissipativity and regularity of the sermigroup ${S_{ε} (t)}_{t ⩾ 0}$ associated to the problem (2.1)–(2.3), mean that the semigroup ${S_{ε} (t)}_{t ⩾ 0}$ associated to the system (2.1)–(2.3), possesses a bounded absorbing set in $ϕ_{3}$ .

Theorem 5.1.
We assume $ϵ < 1$ and that u is the solution of problem ( 2.1 )–( 2.3 ) such that $(u_{0}, u_{1}) \in B_{R_{0}}^{1} (ε) \cap ϕ_{3}$ . Then, the solution u satisfies the following estimate $\begin{matrix} (5.1) & {‖ (u (t), \frac{\partial u (t)}{\partial t}) ‖}_{ϕ_{3}} ⩽ Q (‖ u_{0} ‖_{H^{3}}, ‖ u_{1} ‖_{H^{1}}) e^{- β t} + C, \end{matrix}$ where β and C are the positive constants and Q is a monotonic function.
Proof.
(4.18) can be written in the form $\begin{matrix} (5.2) & \frac{d}{d t} E_{3} (t) + β E_{3} (t) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + c . \end{matrix}$ Applying the Gronwall’s lemma to (5.2), we obtain $\begin{matrix} ‖ u ‖_{H^{3}}^{2} + ε {‖ \frac{\partial u}{\partial t} ‖}_{H^{1}}^{2} ⩽ Q (‖ u_{0} ‖_{H^{3}}, ‖ u_{1} ‖_{H^{1}}) e^{- β t} + C, \end{matrix}$ where β and C are the positive constants and Q is the monotonic function. □
Corollary 5.1.
For $ε < 1$ , the semigroup of operators $S_{ε} (t)$ , $t ⩾ 0$ associated to the system ( 2.1 )–( 2.3 ) is dissipative in $ϕ_{3}$ .

The corrollary is a straightforward consequence of Theorem 5.1.

We denote $\begin{matrix} B_{R_{1}}^{2} (ε) = {(u, \frac{\partial u}{\partial t}) \in ϕ_{3} \cap B_{R_{0}}^{1} (ε) / {‖ (u, \frac{\partial u}{\partial t}) ‖}_{ϕ_{3}} ⩽ R_{1}}, \end{matrix}$ the bounded absorbing set for $S_{ε} (t)$ in phase space $ϕ_{3}$ , where $R_{1}$ is large enough.
6. Existence of global attractor

For $ε < 1$ , we have already proved the dissipativity and régularity of the semigroup ${S_{ε} (t)}_{t ⩾ 0}$ associated to the problem (2.1)–(2.3). It remains to split the semigroup $S_{ε} (t)$ as the sum of two continuous operators, $S_{ε}^{1} (t)$ and $S_{ε}^{2} (t)$ such that the solution u with initial condition belonging to $B_{R_{1}}^{2} (ε) \cap ϕ_{3}$ , can be written as follows $u = v + w$ with $\begin{matrix} S_{ε}^{1} (t) (u (0), \frac{\partial u (0)}{\partial t}) = (v (t), \frac{\partial v (t)}{\partial t}) \end{matrix}$ and $\begin{matrix} S_{ε}^{2} (t) (0, 0) = (w (t), \frac{\partial w (t)}{\partial t}), \end{matrix}$ where $S_{ε}^{1} (t)$ is the solving operator to the linear hyperbolic problem $\begin{array}{l} (6.1) & ε \frac{\partial^{2} v}{\partial t^{2}} + \frac{\partial v}{\partial t} + Δ^{2} v = 0 in [0, T] \times Ω, \\ (6.2) & v = Δ v = 0 on [0, T] \times \partial Ω, \\ (6.3) & v (0, x) = u_{0} (x), \frac{\partial v (0, x)}{\partial t} = u_{1} (x) \forall x \in Ω, \end{array}$ $S_{ε}^{2} (t)$ is the solving operator to the nonlinear hyperbolic problem $\begin{array}{l} (6.4) & ε \frac{\partial^{2} w}{\partial t^{2}} + \frac{\partial w}{\partial t} + Δ^{2} w - Δ f (u) + g (u) = 0 in [0, T] \times Ω, \\ (6.5) & w = Δ w = 0 on [0, T] \times \partial Ω, \\ (6.6) & w (0, x) = 0, \frac{\partial w (0, x)}{\partial t} = 0 \forall x \in Ω, \end{array}$ and to show that the operator $S_{ε}^{1} (t)$ uniformly converges to 0 over all bounded subset of $ϕ_{2}$ and the operator $S_{ε}^{2} (t)$ is regularizing on $ϕ_{3}$ , when the time t tends to the infinity.

We multiply (6.1) by v and integrate over Ω, we obtain $\begin{matrix} (6.7) & \frac{d}{d t} (‖ v ‖^{2} + 2 ε (\frac{\partial v}{\partial t}, v)) + 2 ‖ Δ v ‖^{2} = 2 ε {‖ \frac{\partial v}{\partial t} ‖}^{2} . \end{matrix}$ We multiply (6.1) by $\frac{\partial v}{\partial t}$ and integrate over Ω, we have $\begin{matrix} (6.8) & \frac{d}{d t} (ε {‖ \frac{\partial v}{\partial t} ‖}^{2} + ‖ Δ v ‖^{2}) + 2 ε {‖ \frac{\partial v}{\partial t} ‖}^{2} ⩽ 0 . \end{matrix}$ Summing $ψ_{5}$ (6.7) and (6.8) where $ψ_{5} > 0$ and $1 - ψ_{5} > 0$ , we find $\begin{matrix} (6.9) & \frac{d}{d t} E_{5} (t) + c (‖ Δ v ‖^{2} + ε {‖ \frac{\partial v}{\partial t} ‖}^{2}) ⩽ 0, c > 0, \end{matrix}$ with $\begin{matrix} E_{5} (t) = ‖ Δ v ‖^{2} + ε {‖ \frac{\partial v}{\partial t} ‖}^{2} + ψ_{5} (‖ v ‖^{2} + 2 ε (\frac{\partial v}{\partial t}, v)) . \end{matrix}$ Noting that, we have also $\begin{matrix} (6.10) & c (‖ v ‖_{H^{2}}^{2} + ε {‖ \frac{\partial v}{\partial t} ‖}^{2}) ⩽ E_{5} (t) ⩽ c^{″} (‖ v ‖_{H^{2}}^{2} + ε {‖ \frac{\partial v}{\partial t} ‖}^{2}), c, c^{″} > 0 . \end{matrix}$ We deduce, owing to (6.10) that (6.9) can be written as $\begin{matrix} (6.11) & \frac{d}{d t} E_{5} (t) + β E_{5} (t) ⩽ 0, β > 0 . \end{matrix}$ Applying the Gronwall’s lemma to (6.11), owing to (6.10) we obtain $\begin{matrix} ‖ v ‖_{H^{2}}^{2} + ε {‖ \frac{\partial v}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) e^{- β t} . \end{matrix}$ Thus, $\begin{matrix} (6.12) & {‖ S_{ε} (t) (v_{0}, v_{1}) ‖}_{ϕ_{2}}^{2} = {‖ (v (t), \frac{\partial v (t)}{\partial t}) ‖}_{ϕ_{2}} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) e^{- β t}, \end{matrix}$ where Q is the monotonic function, and $S_{ε}^{1} (t)$ uniformly converges to 0 over all bounded subset of $ϕ_{2}$ , when the time t tends to the infinity.

It remains to prove that the operator $S_{ε}^{2} (t)$ is regularizing on $ϕ_{3}$ , when the time t tends to the infinity.

Frist, we need the uniform estimate of $‖ \nabla Δ w ‖^{2}$ that we shall use in the continuation.

We multiply (6.4) by $\frac{\partial w}{\partial t}$ and integrate over Ω. We find, thanks to (3.15) and (3.19) $\begin{matrix} (6.13) & \frac{d}{d t} (‖ Δ w ‖^{2} + ε {‖ \frac{\partial w}{\partial t} ‖}^{2}) + {‖ \frac{\partial w}{\partial t} ‖}^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ Integrating (6.13) over 0 to t, we obtain $\begin{matrix} ‖ Δ w ‖^{2} + ε {‖ \frac{\partial w}{\partial t} ‖}^{2} + \int_{0}^{t} {‖ \frac{\partial w (s)}{\partial t} ‖}^{2} d s ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ hence $\begin{matrix} (6.14) & ‖ Δ w ‖^{2} ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ We multiply (6.4) by $- Δ \frac{\partial w}{\partial t}$ and integrate over Ω, we obtain $\begin{matrix} (6.15) & \frac{1}{2} \frac{d}{d t} (ε {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} + ‖ \nabla Δ w ‖^{2}) + {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} = (Δ f (u), Δ \frac{\partial w}{\partial t}) + (g (u), Δ \frac{\partial w}{\partial t}) . \end{matrix}$ Noting that $\begin{matrix} ‖ \nabla Δ u ‖ ⩽ R_{1}, \end{matrix}$ and using the continuous embeding of $H^{1} (Ω)$ in $L^{6} (Ω)$ and also (6.12) and (6.14), owing to the fact that $w \in H^{3} (Ω) \cap H_{0}^{1} (Ω)$ , we find $\begin{array}{l} (Δ f (u), Δ \frac{\partial w}{\partial t}) \\ ⩽ ‖ \nabla (Δ f (u)) ‖ ‖ \nabla \frac{\partial w}{\partial t} ‖ \\ ⩽ ‖ \nabla (f^{″} (u) {(\nabla u)}^{2} + f^{'} (u) Δ u) ‖ ‖ \nabla \frac{\partial w}{\partial t} ‖ \\ ⩽ ‖ f^{‴} (u) {(\nabla u)}^{3} + 2 f^{″} (u) \nabla u Δ u + f^{″} (u) \nabla u Δ u + f^{'} (u) \nabla Δ u ‖ ‖ \nabla \frac{\partial w}{\partial t} ‖ \\ ⩽ c ({‖ f^{‴} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla u ‖_{L^{6}}^{6} + {‖ f^{″} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla u ‖_{L^{4}}^{2} ‖ Δ u ‖_{L^{4}}^{2} + {‖ f^{'} (u) ‖}_{L^{\infty}}^{2} ‖ \nabla Δ u ‖^{2}) + \frac{1}{4} {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} \\ ⩽ c (‖ Δ v ‖^{6} + ‖ Δ w ‖^{6} + ‖ Δ v ‖^{2} + ‖ Δ w ‖^{2}) + \frac{1}{4} {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} + C \\ (6.16) & ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) e^{- β t} + \frac{1}{4} {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} + C . \end{array}$ Following (4.15), we have $\begin{matrix} (6.17) & (g (u), Δ \frac{\partial w}{\partial t}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) + \frac{1}{4} {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} . \end{matrix}$ Inserting (6.16) and (6.17) into (6.15) in particular, we have $\begin{matrix} (6.18) & \frac{d}{d t} (‖ \nabla Δ w ‖^{2} + ε {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2}) ⩽ Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) e^{- β t} + C, \end{matrix}$ where β and C are the positive constants and Q is a monotonic function.

Integrating (6.18) over 0 to t, we obtain $\begin{matrix} ‖ w ‖_{H^{3}}^{2} + ε {‖ \nabla \frac{\partial w}{\partial t} ‖}^{2} ⩽ (1 + T^{2}) Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖), \end{matrix}$ which implies $\begin{matrix} (6.19) & {‖ S_{ε}^{2} (t) (0, 0) ‖}_{ϕ_{3}}^{2} ⩽ (1 + T^{2}) Q (‖ u_{0} ‖_{H^{2}}, ‖ u_{1} ‖) . \end{matrix}$ The estimate (6.19) allows to assert that the operator $S_{ε}^{2} (t)$ is regularizing in $ϕ_{3}$ .

We can then deduce the following result.

Theorem 6.1.

For $ε < 1$ , The semigroup $S_{ε} (t)$ possesses the global attractor $A$ who is bounded in $ϕ_{3}$ and compact in $ϕ_{2}$ .

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