On the hyperbolic relaxation of the Cahn–Hilliard equation with a mass source

Abstract

In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results.

Keywords

Cahn-Hilliard equation relaxation term mass source well posedness tumor growth global attractor exponential attractor simulations

1. Introduction

The classical Cahn–Hilliard (CH) equation was proposed in [1,2] in order to describe phase separation processes in binary alloys. During the last decades, it has been analyzed in depth from the mathematical and numerical point of view. It is impossible to quote all the articles related to the mathematical and numerical study of the Cahn–Hilliard equation, the interested ones can see [19]. It is interesting with the Cahn–Hilliard equation in this (or some of its variants) has been used in many other applications, such as deploying in corrosion processes (see [9]), population dynamics (see [5]), tumor growth (see [19]), bacterial films (see [16]), thin films (see [21]), chemistry (see [22]), image processing (see [19]). Recently, researchers have devoted considerable attention to one of its variants which is the hyperbolic relaxation of the Cahn–Hilliard equation. The relaxation term was introduced into the Cahn–Hilliard equation by Galenko et al. (see [6,11–13]), in order to treat more precisely the non-equilibrium effects in the spinodal decomposition generated by fast solidification in some materials (e.g. glasses). It has been studied in [8,14,18,23,24], where the authors proved the existence and uniqueness of the solution, global attractor and a robust family of exponential attractors with appropriate boundary conditions and a regular potential.

We will give a brief description of the relaxed model equations that we will study in this paper with a proliferation term. We assume that an alloy binary is sufficiently cooled to undergo a phase separation process. Naming $u = u (x, t)$ the relative concentration of one component of the system and $J = J (x, t)$ the diffusion flux, the balance law for concentration gives $\begin{matrix} (1.1) & u_{t} = - \nabla \cdot J . \end{matrix}$ In order to describe the evolution for u, we need to introduce a constitutive assumption on J. On case could be $\begin{matrix} (1.2) & τ_{D} J_{t} + J = - \nabla \frac{δ E}{δ u}, \end{matrix}$ where $τ_{D} ⩾ 0$ is the relaxation parameter and $E (u)$ is the total free energy that takes the form as $\begin{matrix} E (u) = \int_{Ω} (\frac{ϵ^{2}}{2} | \nabla u |^{2} + F (u)) d x, \end{matrix}$ where ϵ is a positive constant that measures the interfacial width and $F (u)$ is a double-well potential accounting for the presence of two components (e.g., $F (s) = {(s^{2} - 1)}^{2}$ ). By combining (1.1) with (1.2), and setting $ϵ = 1$ for the sake of simplicity, we obtain $\begin{matrix} (1.3) & τ_{D} u_{t t} + u_{t} + Δ^{2} u - Δ f (u) = 0, \end{matrix}$ where $f (u) = F^{'} (u)$ .

The relaxation term $τ_{D} u_{t t}$ added to the Cahn-Hilliard equation (1.3) changes the type of the equation (from parabolic to hyperbolic) and the analytical properties of its solutions. If $τ_{D} = 0$ , then (1.3) is the classical Cahn-Hilliard equation (see, [1,2,19] and the references therein) for more detail.

Equation (1.3) is usually endowed with boundary and initial conditions on the order parameter, following. $\begin{array}{l} (1.4) & \begin{aligned} u (x, t) = Δ u (x, t) = 0, on \partial Ω \times (0, + \infty), \\ u (0, t) = u_{0} and u_{t} (0, t) = u_{1}, in Ω, \end{aligned} \end{array}$ where $\partial Ω$ is the boundary of the domain Ω occupied by the system.

We consider in this paper the following problem $\begin{matrix} (1.5) & τ_{D} u_{t t} + u_{t} + Δ^{2} u - Δ f (u) = g (u), \end{matrix}$ The g function is used for applications in biology, specifically for tumor growth (see [3,4,10,20]).

This article is organized as follows: in Section 2, we present the problem and fix some notations of operators and spaces. In Section 3, we derive a priori estimates which allow us to prove, in Section 3.1, the existence, uniqueness, and the regularity of solution. In Section 3.2, setting of dissipativity and in Section 4, dissipativity in higher-order spaces. In Section 5, the existence of exponential attractors. Then, we finish in Section 6, with the numeral simulations.

2. Setting of the problem

Let us consider the following initial and boundary value problem in a bounded open interval of $R$ , with boundary $\partial Ω$ : $\begin{array}{l} (2.1) & τ_{D} u_{t t} + u_{t} + Δ^{2} u - Δ f (u) = g (u), in Ω \times (0, + \infty), \\ (2.2) & u (0) = u_{0} and u_{t} (0) = u_{1}, in Ω, \\ (2.3) & u (x, t) = Δ u (x, t) = 0 on \partial Ω \times (0, + \infty), \end{array}$ where u is the order parameter, $τ_{D}$ is the relaxation time of the diffusion flow, f is the nonlinear regular potentials and g is a source term.

We make the following assumptions $\begin{array}{l} (2.4) & f \in C^{2} (R), \\ (2.5) & f (0) = 0, \\ (2.6) & f (s) s ⩾ F (s) - c_{1} ⩾ c_{2}, c_{1}, \forall s \in R, c_{2} ⩾ 0, \\ (2.7) & F (s) ⩾ c_{3} s^{2 k} - c_{4}, c_{3} > 0, c_{4} ⩾ 0, \forall s \in R, k > 0, \\ (2.8) & f^{'} (s) ⩾ - c_{5}, \forall s \in R, c_{5} ⩾ 0, \end{array}$ where $\begin{array}{l} F (s) = \int_{0}^{s} f (t) d t . \end{array}$

The source term g satisfies $\begin{array}{l} (2.9) & g \in C^{1} (R), \\ (2.10) & | g (s) | ⩽ c_{6} s^{k} + c_{7}, c_{6} > 0, c_{7} ⩾ 0, \forall s \in R, k > 0 . \end{array}$

Notations. We denote by $((\cdot, \cdot))$ the usual scalar product on $L^{2} (0, ℓ)$ associated to the norm $‖ \cdot ‖$ . Also, we set $‖ \cdot ‖_{- 1} = ‖ {(- Δ)}^{- \frac{1}{2}} ‖$ , where $- Δ$ stands for the minus Laplace operator associated with Dirichlet boundary conditions. Noting $‖ \cdot ‖_{- 1}$ is equivalent to the usual norm on $H^{- 1} (0, ℓ)$ , where $H^{- 1} (0, ℓ) = H_{0}^{1} {(0, ℓ)}^{'}$ . Besides, $‖ \cdot ‖_{X}$ denotes the norm on the Banach space X. We set $\begin{array}{l} A = - Δ = - \frac{\partial^{2}}{\partial x^{2}} and Ω =] 0, ℓ [\subset R \end{array}$ and $\begin{array}{l} Ψ_{s}^{τ_{D}} = D (A^{\frac{s + 1}{2}}) \times D (A^{\frac{s - 1}{2}}) . \end{array}$

3. A priori estimates

In this section, we first establish some priori estimates of the system (2.1)–(2.3), which are obtained by formal arguments, and prove the existence and uniqueness of the solution, l’existence of a continuous dissipative semigroup, and obtain some regularity who are essential for the continuation.

We rewrite (2.1) in the equivalent form: $\begin{matrix} (3.1) & \frac{d^{2} u}{d t^{2}} + \frac{d u}{d t} + A^{2} u + A f (u) = g (u) . \end{matrix}$ Multiplying (3.1) by $α A^{- 1} u + A^{- 1} u_{t}$ , with $α \in (0, \frac{1}{2 τ_{D}})$ and integrating over Ω, we obtain $\begin{matrix} (3.2) & \frac{d}{d t} Y (t) + (2 - 2 α τ_{D}) ‖ u_{t} ‖_{- 1}^{2} + 2 α ‖ \nabla u ‖^{2} + 2 α \int_{Ω} f (u) u d x = 2 (g (u), α A^{- 1} u + A^{- 1} u_{t}), \end{matrix}$ where $\begin{matrix} (3.3) & Y (t) = 2 α τ_{D} {(u_{t}, u)}_{- 1} + α ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + τ_{D} ‖ u_{t} ‖_{- 1}^{2} + 2 \int_{Ω} F (u) d x . \end{matrix}$ Thanks to Cauchy–Schwarz inequality yields $\begin{array}{l} α τ_{D} | {(u_{t}, u)}_{- 1} | & ⩽ α τ_{D} ‖ u_{t} ‖_{- 1} ‖ u ‖_{- 1} \\ ⩽ \frac{τ_{D}}{4} ‖ u_{t} ‖_{- 1}^{2} + \frac{α}{2} ‖ u ‖_{- 1}^{2} . \end{array}$ Owing to (2.7), we obtain $\begin{array}{l} \int_{Ω} F (u) d x ⩾ c_{2} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} - c_{3} (| Ω |) . \end{array}$ Combinig the above estimates in (3.3), we deduce $\begin{array}{l} c_{7} (τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ‖ u ‖_{L^{2 k} (Ω)}^{2 k}) - c_{3} (| Ω |) \\ ⩽ Y (t) ⩽ c_{8} (τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ‖ u ‖_{L^{2 k} (Ω)}^{2 k}) + c_{3} (| Ω |) . \end{array}$ Thanks to (2.6)–(2.7), we find $\begin{array}{l} α \int_{Ω} f (u) u d x & ⩾ α c_{9} \int_{Ω} F (u) d x - α c_{10} \int_{Ω} d x \\ ⩾ α c_{11} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} - α c_{12} (| Ω |), c_{11}, c_{9} > 0 . \end{array}$ Using (2.10), Young’s and Poincaré inequalities, we get $\begin{array}{l} α | (g (u), A^{- 1} u) | & ⩽ α | g (u) | ‖ A^{- 1} u ‖ \\ ⩽ α c_{13} ‖ u^{k} ‖ ‖ u ‖_{- 1} + α c_{14} ‖ u ‖_{- 1} \\ ⩽ α c_{13} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} + \frac{1}{2} ‖ u ‖_{- 1}^{2} + α c_{15} \end{array}$ and $\begin{array}{l} | (g (u), A^{- 1} u_{t}) | & ⩽ | g (u) | ‖ A^{- 1} u_{t} ‖ \\ ⩽ c_{16} ‖ u^{k} ‖ ‖ u_{t} ‖_{- 1} + c_{17} ‖ u_{t} ‖_{- 1} \\ ⩽ c_{16} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} + \frac{1}{2} ‖ u_{t} ‖_{- 1}^{2} + c_{18} . \end{array}$ Combining the above inequalities, we can rewrite (3.2) as follows $\begin{matrix} (3.4) & \frac{d}{d t} Y (t) + κ Y (t) ⩽ α c_{19}, κ > 0 . \end{matrix}$ Apply the Gronwall’s lemma to (3.4) (see, eg., [19]), we obtain $\begin{matrix} (3.5) & τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ‖ u ‖_{L^{2 k} (Ω)}^{2 k} ⩽ Y (0) exp (- κ t) + c_{20}, κ > 0, c_{20} ⩾ 0 . \end{matrix}$ We multiply (3.1) by $β u + u_{t}$ , with $β \in (0, \frac{1}{2 τ_{D}})$ and integrating over Ω, we have $\begin{array}{l} \frac{d}{d t} H (t) + (2 - 2 β τ_{D}) ‖ u_{t} ‖^{2} + 2 β ‖ A u ‖^{2} + 2 β ((\nabla u) f^{'} (u), \nabla u) + 2 ((\nabla u) f^{'} (u), \nabla u_{t})) \\ (3.6) & = 2 (g (u), β u + u_{t}), \end{array}$ where $\begin{matrix} H (t) = 2 β (u, u_{t}) + β ‖ u ‖^{2} + ‖ A u ‖^{2} + τ_{D} ‖ u_{t} ‖^{2} . \end{matrix}$ Thus, for $β \in (0, \frac{1}{2 τ_{D}})$ , we deduce $\begin{matrix} c_{21} (‖ u ‖^{2} + ‖ A u ‖^{2} + τ_{D} ‖ u_{t} ‖^{2}) ⩽ H (t) ⩽ c_{22} (‖ u ‖^{2} + ‖ A u ‖^{2} + τ_{D} ‖ u_{t} ‖^{2}), c_{21}, c_{22} > 0 . \end{matrix}$ Owing to (2.4)–(2.8), Young’s inequality and $H^{1} (Ω) \subset L^{\infty} (Ω)$ , we have $\begin{array}{l} | (f^{'} (u) \nabla u, \nabla u_{t}) | & ⩽ c_{23} (1 + ‖ u ‖_{L^{\infty} (Ω)}^{2}) ‖ \nabla u ‖ ‖ \nabla u_{t} ‖ \\ ⩽ c_{24} (1 + ‖ \nabla u ‖^{2}) ‖ \nabla u ‖ ‖ \nabla u_{t} ‖ \\ ⩽ c_{24} ‖ \nabla u ‖^{2} + c_{25} ‖ \nabla u ‖_{L^{6} (Ω)}^{6} + \frac{1}{4} ‖ \nabla u_{t} ‖^{2} \end{array}$ and $\begin{matrix} β | (f^{'} (u) \nabla u, \nabla u) | ⩾ - β c_{26} ‖ \nabla u ‖^{2} . \end{matrix}$ Thanks to (2.10), Young’s and Poincaré inequalities, we deduce $\begin{array}{l} β | (g (u), u) | & ⩽ β | g (u) | ‖ u ‖ \\ ⩽ β c_{27} ‖ u^{k} ‖ ‖ \nabla u ‖ + β c_{27} ‖ \nabla u ‖ \\ ⩽ β c_{28} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} + \frac{1}{2} ‖ \nabla u ‖^{2} + α c_{29} \end{array}$ and $\begin{array}{l} | (g (u), u_{t}) | & ⩽ | g (u) | ‖ u_{t} ‖ \\ ⩽ c_{30} ‖ u^{k} ‖ ‖ u_{t} ‖ + c_{31} ‖ u_{t} ‖ \\ ⩽ c_{31} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} + \frac{1}{2} ‖ u_{t} ‖^{2} + c_{32} . \end{array}$ Combining the above estimates, we can rewrite (3.6) as $\begin{array}{l} \frac{d}{d t} H (t) + (1 - 2 β τ_{D}) ‖ u_{t} ‖^{2} + 2 β ‖ A u ‖^{2} + (c_{24} - β c_{33}) ‖ \nabla u ‖^{2} + c_{34} (‖ \nabla u_{t} ‖^{2} + ‖ \nabla u ‖_{L^{6} (Ω)}^{6}) \\ (3.7) & ⩽ β c_{28} ‖ u ‖_{L^{2 k} (Ω)}^{2 k} + β c_{35} . \end{array}$ Thanks to, $1 - 2 β τ_{D} > 0$ , $c_{24} - β c_{33} > 0$ , (3.5), and (3.6), we have $\begin{matrix} (3.8) & \frac{d}{d t} H (t) + κ^{'} H (t) + c_{36} (‖ \nabla u ‖^{2} + ‖ \nabla u ‖_{L^{6} (Ω)}^{6} + ‖ \nabla u_{t} ‖^{2}) ⩽ β c_{37} . \end{matrix}$ Apply the Gronwall’s lemma to (3.8), we conclude $\begin{matrix} (3.9) & H (t) + \int_{0}^{T} (‖ \nabla u ‖^{2} + ‖ \nabla u ‖_{L^{6} (Ω)}^{6} + ‖ \nabla u_{t} ‖^{2}) d s ⩽ exp (- κ^{'} t) H (0) + c_{38}, κ > 0, c_{38} ⩾ 0 . \end{matrix}$

3.1. Existence and uniqueness of solutions

The estimates below allow us to state the following theorem.

Theorem 3.1.
For $(u_{0}, u_{1}) \in Ψ_{0}^{τ_{D}}$ . Assuming ( 2.4 )–( 2.8 ) and ( 2.10 ) are holds. Then there exists a solution $(u, u_{t})$ of system ( 2.1 )–( 2.3 ) such that, $\begin{array}{l} (u, u_{t}) \in L^{\infty} (R^{+}, H_{0}^{1} (Ω) \times H^{- 1} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω) \times L^{2} (Ω)) for all T > 0 . \end{array}$
Proof.
The proof of the existence is based on the Galerkin standard method.

Consider a spectral basis ${(ω_{i})}_{i ⩾ 1}$ of $- Δ$ associated with eigenvalues, $0 < λ_{1} < λ_{2} < \dots < \dots$ which forms an orthonormal basis in $L^{2} (Ω)$ and orthogonal in $H_{0}^{1} (Ω)$ . We pose $V_{n} = span {ω_{1}, \dots, ω_{n}}$ this spectral basis, and $\begin{array}{l} u_{n} = \sum_{i = 1}^{n} u_{i, n} ω_{i} . \end{array}$ Consider the following approximating problem, written in the functional form $\begin{array}{l} (3.10) & τ_{D} \frac{d^{2} u_{n}}{d t^{2}} + \frac{d u_{n}}{d t} + A^{2} u_{n} + A f (u_{n}) - g (u_{n}) = 0 \\ (3.11) & u_{n} (0) = u_{0, n} and \frac{d u_{n} (0)}{d t} = u_{1, n} . \end{array}$ Replacing u by $u_{n}$ in estimates (3.5) and (3.9), we conclude that $\begin{array}{l} u_{n} is bounded in L^{\infty} (R^{+}, H_{0}^{1} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω)) \\ \frac{d u_{n}}{d t} is bounded in L^{\infty} (R^{+}, H^{- 1} (Ω)) \cap L^{2} (0, T; L^{2} (Ω)) . \end{array}$ Finally, the passage to the limit is based on classical (Aubin–Lions type) compactness results, we find the result. □
Theorem 3.2.
Under the assumption of Theorem 3.1 and for $k ⩾ 1$ , the solution $(u, u_{t})$ is unique.
Proof.
Consider $(u_{1}, \frac{d u_{1}}{d t})$ and $(u_{2}, \frac{d u_{2}}{d t})$ two solutions of systems (2.1)–(2.3) with respective initial conditions $(u_{0, 1}, u_{1, 1})$ and $(u_{0, 2}, u_{1, 2})$ in $Ψ_{0}^{τ_{D}}$ . we pose: $\begin{array}{l} (u, u_{t}) = (u_{1} - u_{2}, \frac{d u_{1}}{d t} - \frac{d u_{2}}{d t}) \end{array}$ The system (2.1)–(2.3) becomes $\begin{array}{l} (3.12) & τ_{D} u_{t t} + u_{t} + A^{2} u + A (f (u_{1}) - f (u_{2})) - (g (u_{1}) - g (u_{2})) = 0, \\ (3.13) & u (0) = u_{0, 1} - u_{0, 2} and u_{t} (0) = u_{1, 1} - u_{1, 2} . \end{array}$ Multiplying (3.12) by $λ A^{- 1} u + A^{- 1} u_{t}$ , with $λ \in (0, \frac{1}{2 τ_{D}})$ and integrating over Ω, we have $\begin{array}{l} \frac{d}{d t} Γ (t) + (2 - 2 λ τ_{D}) ‖ u_{t} ‖_{- 1}^{2} + 2 λ ‖ \nabla u ‖^{2} + 2 (f (u_{1}) - f (u_{2}), λ u + u_{t}) \\ (3.14) & - 2 (g (u_{1}) - g (u_{2}), λ A^{- 1} u + A^{- 1} u_{t}) = 0, \end{array}$ where $\begin{matrix} Γ (t) = 2 λ τ_{D} {(u_{t}, u)}_{- 1} + λ ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + τ_{D} ‖ u_{t} ‖_{- 1}^{2} . \end{matrix}$ Thanks to Cauchy–Schwarz inequality yields $\begin{matrix} c_{39} (‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + τ_{D} ‖ u_{t} ‖_{- 1}^{2}) ⩽ Γ (t) ⩽ c_{40} (‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + τ_{D} ‖ u_{t} ‖_{- 1}^{2}), c_{39}, c_{40} > 0 . \end{matrix}$ Owing to (2.4)–(2.8), we get $\begin{array}{l} λ | (f (u_{1}) - f (u_{2}), u) | & = λ | (u \int_{0}^{1} f^{'} (s u_{1} + (1 - s) u_{2}) d s, u) | \\ ⩾ - λ c_{4} ‖ u ‖^{2} \\ ⩾ - λ c_{41} ‖ u ‖_{- 1}^{2} \end{array}$ and $\begin{array}{l} | \nabla (f (u_{1}) - f (u_{2}) | ⩽ & ‖ \nabla (u \int_{0}^{1} f^{'} (s u_{1} + (1 - s) u_{2}) d s) ‖ \\ ⩽ & ‖ \nabla u (\int_{0}^{1} f^{'} (s u_{1} + (1 - s) u_{2}) d s) ‖ \\ + ‖ u (\int_{0}^{1} f^{″} (s u_{1} + (1 - s) u_{2}) (s \nabla u_{1} + (1 - s) \nabla u_{2}) d s) ‖ \\ ⩽ & Q (‖ u_{0, 1} ‖_{H^{1} (Ω)}, ‖ u_{0, 2} ‖_{H^{1} (Ω)}) (‖ \nabla u ‖ + ‖ u | \nabla u_{1} | ‖ + ‖ u | \nabla u_{2} | ‖) \\ ⩽ & Q (‖ u_{0, 1} ‖_{H^{1} (Ω)}, ‖ u_{0, 2} ‖_{H^{1} (Ω)}) ‖ \nabla u ‖ . \end{array}$ Thanks to Young’s inequality, we deduce $\begin{array}{l} ‖ \nabla (f (u_{1}) - f (u_{2}) ‖ ‖ u_{t} ‖_{- 1} ⩽ Q (‖ u_{0, 1} ‖_{H^{1} (Ω)}, ‖ u_{0, 2} ‖_{H^{1} (Ω)}) ‖ \nabla u ‖^{2} + \frac{1}{4} ‖ u_{t} ‖_{- 1}^{2} \end{array}$ Owing to (2.9), Young’s and Poincaré inequalities, we obtain $\begin{array}{l} λ | (g (u_{1}) - g (u_{2}), A^{- 1} u) | & ⩽ λ ‖ (u \int_{0}^{1} g^{'} (s u_{1} + (1 - s) u_{2}) d s ‖ ‖ u ‖_{- 1} \\ ⩽ Q (λ, ‖ u_{0, 1} ‖_{H^{1}}^{k - 1}, ‖ u_{0, 2} ‖_{H^{1}}^{k - 1}) ‖ \nabla u ‖^{2} + \frac{1}{2} ‖ u ‖_{- 1}^{2} \end{array}$ and $\begin{array}{l} | (g (u_{1}) - g (u_{2}), A^{- 1} u_{t}) | & ⩽ ‖ (u \int_{0}^{1} g^{'} (s u_{1} + (1 - s) u_{2}) d s ‖ ‖ u_{t} ‖_{- 1} \\ ⩽ Q (‖ u_{0, 1} ‖_{H^{1}}^{k - 1}, ‖ u_{0, 2} ‖_{H^{1}}^{k - 1}) ‖ \nabla u ‖^{2} + \frac{1}{2} ‖ u_{t} ‖_{- 1}^{2} . \end{array}$ Combining the above estimates in (3.14), we have $\begin{matrix} (3.15) & \frac{d}{d t} Γ (t) ⩽ Q (λ, ‖ u_{0, 1} ‖_{H^{1} (Ω)}, ‖ u_{0, 2} ‖_{H^{1} (Ω)}, ‖ u_{0, 1} ‖_{H^{1}}^{k - 1}, ‖ u_{0, 2} ‖_{H^{1}}^{k - 1}) Γ (t) . \end{matrix}$ Apply the Gronwall’s lemma to (3.15), we conclude $\begin{matrix} (3.16) & {‖ (u, u_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} ⩽ e^{Q t} {‖ (u (0), u_{t} (0)) ‖}_{Ψ_{0}^{τ_{D}}}^{2} . \end{matrix}$ Hence the uniqueness, as well as the continuous dependence with respect to the initial data. □
Remark 3.3.
For $k \in] 0, 1 [$ , we do not have the uniqueness of the solution for the system (2.1)–(2.3).

3.2. Dissipativity

Thanks to Theorem 3.2, we can state the following theorem

Theorem 3.4.
Under the assumption of Theorem 3.2 , then the system ( 2.1 )–( 2.3 ) define a continuous semigroup $\begin{array}{l} S_{τ_{D}} (t) : Ψ_{0}^{τ_{D}} \to Ψ_{0}^{τ_{D}}, \end{array}$ by the expression $\begin{array}{l} S_{τ_{D}} (t) (u_{0}, u_{1}) = (u (t), u_{t} (t)), \end{array}$ where $(u (t), u_{t} (t))$ is the solution to ( 2.1 )–( 2.3 ).
Corollary 3.5.
Under the assumption of Theorem 3.1 , the semigroup ${S_{τ_{D}} (t), t ⩾ 0}$ is dissipative in $Ψ_{0}^{τ_{D}}$ .
Proof.
The proof of Corollary follows directly from the estimate (3.5). □
Remark 3.6.
We can assume, without loss of generality, that $B_{0}$ is positively invariant to ${S_{τ_{D}} (t), t ⩾ 0}$ , i.e., $\begin{array}{l} S_{τ_{D}} (t) B_{0} \subset B_{0}, \forall t ⩾ 0 . \end{array}$
Lemma 3.7.
For any $R ⩾ 0$ and $k ⩾ 1$ , there exist $Q = Q (R) ⩾ 0$ such that, for any two initial data $u_{0}, u_{1} \in Ψ_{0}^{τ_{D}}$ with $‖ u_{i} ‖_{Ψ_{0}^{τ_{D}}} ⩽ R$ , there holds $\begin{matrix} (3.17) & {‖ S_{τ_{D}} (t) u_{0} - S_{τ_{D}} (t) u_{1} ‖}_{Ψ_{0}^{τ_{D}}} ⩽ e^{K t} ‖ u_{0} - u_{1} ‖_{Ψ_{0}^{τ_{D}}}, \end{matrix}$ where $u_{0} = (u_{0}^{1}, u_{0}^{2})$ and $u_{1} = (u_{1}^{1}, u_{1}^{2})$ .
Proof.
Given two solutions $u^{1}$ and $u^{2}$ corresponding to different initial data $u_{0}, u_{1} \in Ψ_{0}^{τ_{D}}$ , the difference $u = u^{1} - u^{2}$ fulfills $\begin{matrix} (3.18) & τ_{D} u_{t t} + u_{t} + A^{2} u + A (f (u^{1}) - f (u^{2})) - (g (u^{1}) - g (u^{2})) = 0 . \end{matrix}$ Multiplying (3.18) by $A^{- 1} u_{t}$ , we have $\begin{matrix} (3.19) & \frac{d}{d t} (τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2}) + 2 ‖ u_{t} ‖_{- 1}^{2} = - 2 (f (u^{1}) - f (u^{2}), u_{t}) + 2 (g (u^{1}) - g (u^{2}), A^{- 1} u_{t}) . \end{matrix}$ Thanks to Theorem 3.2, we deduce $\begin{matrix} (3.20) & \frac{d}{d t} (τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2}) + 2 ‖ u_{t} ‖_{- 1}^{2} ⩽ c (τ_{D} ‖ u_{t} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2}), c > 0 . \end{matrix}$ Applied the Gronwall’s lemma to 3.20, we obtain the result of Lemma. □
Theorem 3.8.
Under the assumption of Theorem 3.2 , and for, $(u_{0}, u_{1}) \in (H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times L^{2} (Ω)$ then the system ( 2.1 )–( 2.3 ) has a unique solution $(u, u_{t})$ such that $\begin{array}{l} (u, u_{t}) \in L^{\infty} (R^{+}, H^{2} (Ω) \cap H_{0}^{1} (Ω) \times L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)), for all T > 0 \end{array}$
Proof.
Thanks to estimates (3.5), (3.9), we find the result of Theorem 3.8. □
Theorem 3.9.
Under the hypotheses of Theorems 3.1 and 3.2 , the Semigroup ${S_{τ_{D}} (t), t ⩾ 0}$ defined on the phase $Ψ_{0}^{τ_{D}}$ has a global attractor $A_{τ_{D}}$ in $Ψ_{1}^{τ_{D}}$ .
Proof.
We decompose the semigroup ${S_{τ_{D}} (t), t ⩾ 0}$ into a sum of two semigroups so that the first tends to 0 when t tends to infinity and the second is asymptotically compact to $Ψ_{1}^{τ_{D}}$ $\begin{array}{l} S_{τ_{D}} (t) = S_{τ_{D}}^{1} (t) + S_{τ_{D}}^{2} (t), t ⩾ 0 . \end{array}$ Consider the following decomposition $\begin{matrix} (3.21) & (u (t), u_{t} (t)) = (w (t), w_{t} (t)) + (v (t), v_{t} (t)), \end{matrix}$ where $(v (t), v_{t} (t))$ is the solution of the system $\begin{array}{l} (3.22) & τ_{D} v_{t t} + v_{t} + A^{2} v + A (f (u) - f (w)) + v = g (u) - g (w), \\ (3.23) & v (0) = u_{0}, v_{t} (0) = u_{1}, \end{array}$ associated with the semigroup { $S_{τ_{D}}^{1} (t)$ , $t ⩾ 0$ } and $(w (t), w_{t} (t))$ is the solution of the system $\begin{array}{l} (3.24) & τ_{D} w_{t t} + w_{t} + A^{2} w + A f (w) + w - u = g (w), \\ (3.25) & w (0) = 0, w_{t} (0) = 0, \end{array}$ associated with the semigroup { $S_{τ_{D}}^{2} (t)$ , $t ⩾ 0$ }.

To continue the proof, we need the following lemma. Lemma 3.10.
Under the hypotheses of Theorem 3.1 , the semigroup ${S_{τ_{D}}^{2} (t), t ⩾ 0}$ associated with system ( 3.24 )–( 3.25 ) is dissipative in $Ψ_{1}^{τ_{D}}$ .
Proof.
We multiply (3.24) by $β w + w_{t}$ and integrating over Ω, we obtain $\begin{array}{l} \frac{d}{d t} Γ (t) + (2 - 2 β τ_{D}) ‖ w_{t} ‖^{2} + 2 β ‖ A w ‖^{2} + 2 β ‖ w ‖^{2} + 2 β ((\nabla w) f^{'} (w), \nabla w) \\ + 2 ((\nabla w) f^{'} (w), \nabla w_{t})) \\ (3.26) & = 2 (w_{t}, u) + 2 β (w, u) + 2 (g (w), β w + w_{t}) \end{array}$ where $\begin{matrix} Γ (t) = 2 β (w, w_{t}) + β ‖ w ‖^{2} + ‖ A w ‖^{2} + τ_{D} ‖ w_{t} ‖^{2} + ‖ w ‖^{2} . \end{matrix}$ Thanks to Theorem 3.8 and Holder’s inequality, we find $\begin{array}{l} β | (w, u) | + | (w_{t}, u) | & ⩽ β ‖ w ‖ ‖ u ‖ + ‖ w_{t} ‖_{- 1} ‖ \nabla u ‖ \\ ⩽ β c ‖ w ‖^{2} + \frac{1}{4} ‖ w_{t} ‖_{- 1}^{2} + β c^{'} . \end{array}$ Combining the above of expression, we deduce $\begin{matrix} c_{5} (‖ w ‖^{2} + ‖ A w ‖^{2} + τ_{D} ‖ w_{t} ‖^{2}) ⩽ Γ (t) ⩽ c_{6} (‖ w ‖^{2} + ‖ A w ‖^{2} + τ_{D} ‖ w_{t} ‖^{2}) . \end{matrix}$ Using the steps in the a priori estimation section, we have $\begin{matrix} (3.27) & \frac{d}{d t} Γ (t) + c Γ (t) ⩽ c^{'}, c > 0, c^{'} ⩾ 0 . \end{matrix}$ Applied the Gronwall’s to (3.27) and thanks to (3.25), we get $\begin{matrix} (3.28) & {‖ (w (t), w_{t} (t)) ‖}_{Ψ_{1}^{τ_{D}}}^{2} ⩽ c^{″} . \end{matrix}$ □

Multiplying (3.22) by $λ A^{- 1} v + A^{- 1} v_{t}$ and integrating over Ω, we deduce $\begin{array}{l} \frac{d}{d t} Γ_{1} (t) + (2 - 2 λ τ_{D}) ‖ v_{t} ‖_{- 1}^{2} + λ ‖ v ‖_{- 1}^{2} + 2 λ ‖ \nabla v ‖^{2} + 2 (f (u) - f (w), λ v + v_{t}) \\ (3.29) & = 2 (g (u) - g (w), λ A^{- 1} v + A^{- 1} v_{t}), \end{array}$ where $\begin{matrix} Γ_{1} (t) = 2 λ τ_{D} {(v, v_{t})}_{- 1} + τ_{D} ‖ v_{t} ‖_{- 1}^{2} + λ ‖ v ‖_{- 1}^{2} + ‖ \nabla v ‖^{2} + ‖ v ‖_{- 1}^{2} . \end{matrix}$ Thus, for $λ \in (0, \frac{1}{2 τ_{D}})$ small enough, we obtain $\begin{matrix} (3.30) & c_{5} (τ_{D} ‖ v_{t} ‖_{- 1}^{2} + ‖ v ‖_{- 1}^{2} + ‖ \nabla v ‖^{2}) ⩽ Γ_{1} (t) ⩽ c_{6} (τ_{D} ‖ v_{t} ‖_{- 1}^{2} + ‖ v ‖_{- 1}^{2} + ‖ \nabla v ‖^{2}), c_{5}, c_{6} > 0 . \end{matrix}$ Owing to (2.4)–(2.8), Young’s and Hölder inequalities, we conclude $\begin{array}{l} λ | (f (u) - f (w), v) | & = λ | (v \int_{0}^{1} f^{'} (s u + (1 - s) w) d s, v) | \\ ⩾ - c_{7} λ ‖ \nabla v ‖^{2} - \frac{1}{2} ‖ v ‖_{- 1}^{2} \end{array}$ and $\begin{matrix} | (A^{\frac{1}{2}} (f (u) - f (w)), A^{- \frac{1}{2}} v_{t}) | ⩽ & ‖ \nabla (v \int_{0}^{1} f^{'} (s u + (1 - s) w) d s) ‖ ‖ v_{t} ‖_{- 1} \\ ⩽ & ‖ v \int_{0}^{1} f^{″} (s u + (1 - s) w) (s \nabla u + (1 - s) \nabla w) d s ‖ ‖ v_{t} ‖_{- 1} \\ + ‖ \nabla v \int_{0}^{1} f^{'} (s u + (1 - s) w) d s ‖ ‖ v_{t} ‖_{- 1} \\ ⩽ & c ‖ \nabla v ‖ ‖ v_{t} ‖_{- 1} + c^{'} (‖ u ‖_{H^{2} (Ω)} + ‖ w ‖_{H^{2} (Ω)})) ‖ \nabla v ‖ ‖ v_{t} ‖_{- 1} . \end{matrix}$ Thanks to Lemma 3.10, Theorem 3.8 and Young’s inequality, we have $\begin{matrix} | (A^{\frac{1}{2}} (f (u) - f (w)), A^{- \frac{1}{2}} v_{t}) | ⩽ c^{″} ‖ \nabla v ‖^{2} + \frac{1}{2} ‖ v_{t} ‖_{- 1}^{2} . \end{matrix}$ Combining the above estimates, we can rewrite (3.29) as follows $\begin{matrix} (3.31) & \frac{d}{d t} Γ_{1} (t) + κ Γ_{t} (t) ⩽ 0 . \end{matrix}$ Apply the Gronwall’s lemma to (3.31), we conclude $\begin{matrix} (3.32) & {‖ (v, v_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} ⩽ e^{- κ t} {‖ (u_{0}, u_{1}) ‖}_{Ψ_{0}^{τ_{D}}}^{2}, κ > 0 . \end{matrix}$ Thus, the first part of Theorem 3.9 is proved.

The proof of the second part of the Theorem follows directly from the Lemma 3.10. □
Corollary 3.11.
The global attractor $A_{τ_{D}}$ is bounded in $Ψ_{1}^{τ_{D}}$ , with a bound independent of $τ_{D}$ .

4. Dissipativity in higher-order spaces

Our goal in this part is to show that for f and g satisfying compatibility conditions, the semigroup ${S_{τ_{D}} (t), t ⩾ 0}$ defined on the space $Ψ_{n}^{τ_{D}}$ (for $1 ⩽ n ⩽ 4$ ) has an absorbing and closed set $B_{τ_{D}}^{n}$ in $Ψ_{n}^{τ_{D}}$ . Assume that $\begin{array}{l} (4.1) & f \in C^{n + 1} (R) and f^{″} (0) = 0 for n = 3, 4, \end{array}$ and $\begin{matrix} (4.2) & g \in C^{n} (R) and g^{(n)} (0) = 0 if k > n . \end{matrix}$ If $n = 1$ , then (4.1)–(4.2) does not add anything to our previous assumptions on f and g. Notice that the derivative of the classical Cahn–Hilliard potential satisfies (4.1) for every $n \in N$ .

Theorem 4.1.
Let ( 4.1 )–( 4.2 ) some hold for $1 ⩽ n ⩽ 4$ . Then there exist $R_{n} > 0$ such that the closed ball $B_{τ_{D}}^{n}$ of $Ψ_{n}^{τ_{D}}$ centered at zero of radius $R_{n}$ is an absorbing set for ${S_{τ_{D}} (t), t ⩾ 0}$ in $Ψ_{n}^{τ_{D}}$ . That is, for every bounded set $B \subset Ψ_{n}^{τ_{D}}$ , there exist $t_{n} = t_{n} (B)$ such that $\begin{array}{l} S_{τ_{D}} (t) B \subset B_{τ_{D}}^{n}, t ⩾ t_{n} . \end{array}$
Proof.
The proof of the Theorem follows directly from the proof of the lemma. Lemma 4.2.
Let the hypotheses of Theorem 4.1 hold. Given $ρ_{n - 1}, ρ_{n} ⩾ 0$ , there are

$K_{n - 1} = K_{n - 1} (ρ_{n - 1}) ⩾ 0$ $K_{n} = K_{n} (ρ_{n}) ⩾ 0$ and $ν_{n} > 0$ such that, if $\begin{array}{l} {‖ (u_{0}, u_{1}) ‖}_{Ψ_{n - 1}^{τ_{D}}} ⩽ ρ_{n - 1} and {‖ (u_{0}, u_{1}) ‖}_{Ψ_{n}^{τ_{D}}} ⩽ ρ_{n}, \end{array}$ the following inequality holds $\begin{array}{l} {‖ S_{τ_{D}} (t) (u_{0}, u_{1}) ‖}_{Ψ_{n}^{τ_{D}}} ⩽ K_{n} e^{- ν_{n} t} + K_{n - 1} . \end{array}$
Proof.
In this proof, from $c_{j} ⩾ 0$ depended on $ρ_{n - 1}$ . We are going to reason by recurrent on n. When $n = 1$ it is easy to conclude that (3.5). Hence, assuming that the result holds for $n - 1$ , we have $\begin{matrix} (4.3) & sup_{‖ (u_{0}, u_{1}) ‖_{Ψ_{n - 1}^{τ_{D}}} ⩽ ρ_{n - 1}} ({‖ A^{\frac{n}{2}} u ‖}^{2} + ‖ u ‖_{L^{2 k - 2 (n - 1)} (Ω)}^{2 k - 2 (n - 1)} + τ_{D} {‖ A^{\frac{n - 2}{2}} u_{t} ‖}^{2}) ⩽ c . \end{matrix}$ In particular, the assumptions to f yields $\begin{matrix} (4.4) & sup_{t ⩾ 0} (\sum_{i = 0}^{n + 1} {‖ f^{(i)} (u) ‖}_{L^{\infty} (Ω)}) ⩽ c_{1} . \end{matrix}$ Thanks to formula the derivative of a composition of functions of the Faa di Bruno, there is $\begin{array}{l} (4.5) & D^{n} f (u) = f^{'} (u) D^{n} u + Γ \\ (4.6) & D^{n} g (u) = g^{'} (u) D^{n} u + Γ_{1}, \end{array}$ with $Γ = Γ_{1} = 0$ if $n = 1$ , and $\begin{array}{l} Γ = \sum_{k = 2}^{n + 1} Γ_{k} f^{(k)} (u) \end{array}$ $\begin{array}{l} Γ_{1} = \sum_{k = 2}^{n + 1} Γ_{k} g^{(k)} (u) \end{array}$ where $Γ_{k}$ is a linear combination of terms of the for $\begin{array}{l} {(D^{1} u)}^{i_{1}} {(D^{2} u)}^{i_{2}} \dots {(D^{n - 1} u)}^{i_{n - 1}}, \end{array}$ for some non-negative integers $i_{1}, \dots, i_{n - 1}$ satisfying $\begin{array}{l} i_{1} + \dots + i_{n - 1} = k and 2 i_{1} + \dots + (n - 1) i_{n - 1} = n . \end{array}$ Multiplying (3.1) by $α A^{n - 1} u + A^{n - 1} u_{t}$ , we obtain $\begin{array}{l} \frac{d}{d t} Y (t) + (2 - 2 α τ_{D}) {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + 2 α {‖ A^{\frac{n + 1}{2}} u ‖}^{2} + \int_{0}^{ℓ} u_{t} g^{″} (u) {| A^{\frac{n - 1}{2}} u |}^{2} d x \\ = - 2 α (f^{'} (u) A^{\frac{n}{2}} u + Γ, A^{\frac{n}{2}} u) + 2 α (g^{'} (u) A^{\frac{n - 1}{2}} u + Γ_{1}, A^{\frac{n - 1}{2}} u) \\ (4.7) & + \int_{0}^{ℓ} u_{t} f^{″} (u) {| A^{\frac{n - 1}{2}} u |}^{2} d x - 2 (Γ, A^{\frac{n}{2}} u_{t}) + 2 (Γ_{1}, A^{\frac{n - 1}{2}} u_{t}), \end{array}$ where $\begin{array}{l} Y (t) = & 2 α τ_{D} (A^{\frac{n - 1}{2}} u_{t}, A^{\frac{n - 1}{2}} u) + {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + {‖ A^{\frac{n + 1}{2}} u ‖}^{2} \\ (4.8) & + \int_{0}^{ℓ} {| A^{\frac{n}{2}} u |}^{2} f^{'} (u) d x - \int_{0}^{ℓ} {| A^{\frac{n - 1}{2}} u |}^{2} g^{'} (u) d x . \end{array}$ Thus, for $α \in (0, \frac{1}{2 τ_{D}})$ we deduce $\begin{matrix} 2 α τ_{D} | (A^{\frac{n - 1}{2}} u_{t}, A^{\frac{n - 1}{2}} u) | ⩽ τ_{D} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + \frac{α}{2} {‖ A^{\frac{n - 1}{2}} u ‖}^{2} . \end{matrix}$ Thanks to (4.3) and Young’s inequality, we obtain $\begin{array}{l} - \int_{0}^{ℓ} {| A^{\frac{n - 1}{2}} u |}^{2} g^{'} (u) d x & ⩽ c_{2} ‖ g^{'} (u) ‖ \\ ⩽ ‖ u ‖_{L^{2 k - 2} (Ω)}^{2 k - 2} + c_{3} \end{array}$ and $\begin{array}{l} \int_{0}^{ℓ} {| A^{\frac{n}{2}} u |}^{2} f^{'} (u) d x ⩽ c_{4} . \end{array}$ Combining the above estimates in (4.8), find $\begin{array}{l} c_{5} (τ_{D} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + {‖ A^{\frac{n + 1}{2}} u ‖}^{2} + ‖ u ‖_{L^{2 k - 2} (Ω)}^{2 k - 2}) - c_{6} \\ ⩽ Y (t) ⩽ c_{7} (τ_{D} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + {‖ A^{\frac{n + 1}{2}} u ‖}^{2} + ‖ u ‖_{L^{2 k - 2} (Ω)}^{2 k - 2}) + c_{8} . \end{array}$ Thanks to (4.3), Young’s inequality, we obtain $\begin{array}{l} \int_{0}^{ℓ} u_{t} f^{″} (u) {| A^{\frac{n - 1}{2}} u |}^{2} d x & ⩽ c_{9} ‖ u_{t} ‖ {‖ f^{″} (u) ‖}_{L^{\infty} (Ω)} \\ ⩽ c_{10} ‖ A^{\frac{n - 1}{2}} u_{t} ‖ \\ ⩽ \frac{1}{2} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + c_{11} \end{array}$ and $\begin{array}{l} \int_{0}^{ℓ} u_{t} g^{″} (u) {| A^{\frac{n - 1}{2}} u |}^{2} d x & ⩽ c_{12} ‖ u_{t} ‖ ‖ g^{″} (u) ‖ \\ ⩽ \frac{1}{2} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + c_{13} ‖ u ‖_{L^{2 k - 4} (Ω)}^{2 k - 4} . \end{array}$ Owing to (4.3)–(4.6), we obtain $\begin{array}{l} 2 α | (Γ_{1}, A^{\frac{n - 1}{2}} u) | ⩽ α c_{14} \end{array}$ and $\begin{array}{l} 2 | (Γ_{1}, A^{\frac{n - 1}{2}} u_{t}) | ⩽ c_{15} . \end{array}$ Moreover $\begin{array}{l} 2 α | (g^{'} (u) A^{\frac{n - 1}{2}} u, A^{\frac{n - 1}{2}} u) | & ⩽ c α ‖ g^{'} (u) ‖ {‖ A^{\frac{n - 1}{2}} u ‖}^{2} \\ ⩽ c_{20} ‖ u ‖_{L^{2 k - 2} (Ω)}^{2 k - 2} + α c_{21} . \end{array}$ Using (4.3)–(4.5), Young’s inequality, we deduce $\begin{array}{l} - 2 α | (Γ, A^{\frac{n}{2}} u) | ⩽ α c_{16} \end{array}$ and $\begin{array}{l} - 2 | (Γ, A^{\frac{n}{2}} u_{t}) | & = c_{17} | (A^{\frac{1}{2}} Γ, A^{\frac{n - 1}{2}} u_{t}) | \\ ⩽ \frac{1}{2} {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + c_{18} . \end{array}$ Finally, collecting the above estimates, we conclude $\begin{matrix} (4.9) & \frac{d}{d t} Y (t) + (\frac{3}{2} - 2 α τ_{D}) {‖ A^{\frac{n - 1}{2}} u_{t} ‖}^{2} + 2 α {‖ A^{\frac{n + 1}{2}} u ‖}^{2} + c_{23} ‖ u ‖_{L^{2 k - 4} (Ω)}^{2 k - 4} ⩽ c_{20} Y (t) + c^{'} . \end{matrix}$ Thus, for $\frac{3}{2} - 2 α τ_{D} > 0$ , we deduce $\begin{matrix} (4.10) & \frac{d}{d t} Y (t) + 2 κ Y (t) + ⩽ c Y (t) + c^{'}, κ > 0, c > 0, c^{'} ⩾ 0 . \end{matrix}$ Applying the Lemma 2.1 of [7] to (4.10), we conclude $\begin{matrix} Y (t) ⩽ c^{″} e^{- κ t} Y (0) + c^{‴}, κ > 0, c^{″} > 0, c^{‴} ⩾ 0 . \end{matrix}$ Hence, $\begin{matrix} (4.11) & {‖ (u, u_{t}) ‖}_{Ψ_{n}^{τ_{D}}} ⩽ c . \end{matrix}$ □

□

Actually, up to (possibly) enlarging the radius $R_{n}$ , the absorbing set $B_{τ_{D}}^{n}$ is exponentially attracting in $Ψ_{0}^{τ_{D}}$ as well.
Theorem 4.3.
Consider the hypotheses of Theorem 4.1 for $1 ⩽ n ⩽ 4$ hold. Then there exist $ν_{n} > 0$ and a positive increasing function $J_{n}$ such that, for any bounded set $B \subset Ψ_{0}^{τ_{D}}$ , $\begin{array}{l} {dist}_{Ψ_{0}^{τ_{D}}} (S_{τ_{D}} (t) B, B_{τ_{D}}^{n}) ⩽ J_{n} (R) e^{- ν_{n} t}, \forall t ⩾ 0, \end{array}$ where $R = {sup}_{(u_{0}, u_{1}) \in B} ‖ (u_{0}, u_{1}) ‖_{Ψ_{0}^{τ_{D}}}$ .
Proof.
In the proof, we will use the following two lemmas. Lemma 4.4.
Let $S (t)$ be a strongly continuous semigroup on a Banach space Ψ. Let $B_{0}, B_{1}, B_{3} \subset Ψ$ and satisfy that $\begin{array}{l} {dist}_{Ψ} (S (t) B_{0}, B_{1}) ⩽ K_{1} e^{- ν_{1} t}, {dist}_{Ψ} (S (t) B_{1}, B_{2}) ⩽ K_{2} e^{- ν_{2} t}, \end{array}$ for some $ν_{1}, ν_{2} > 0$ and $K_{1}, K_{2} ⩾ 0$ . Assume that, for $z_{1}, z_{2} \in Ψ$ , then hold $\begin{array}{l} {‖ S (t) z_{1} - S (t) z_{2} ‖}_{Ψ} ⩽ e^{ν_{0} t} ‖ z_{1} - z_{2} ‖_{Ψ}, \end{array}$ for some $ν_{0} ⩾ 0$ . Then it follows that $\begin{array}{l} {dist}_{Ψ} (S (t) B_{0}, B_{2}) ⩽ (K_{1} + K_{2}) e^{- ν t}, \end{array}$ where $ν = \frac{ν_{1} ν_{2}}{ν_{0} + ν_{1} + ν_{2}}$ . Here, ${dist}_{Ψ}$ denotes the usual Hausdorff semidistance in Ψ.
Lemma 4.5.
Let ( 4.1 )–( 4.2 ) some hold for $1 ⩽ n ⩽ 4$ . Then, up to toking a possibly larger $R_{n}$ , we have $\begin{array}{l} {dist}_{Ψ_{0}^{τ_{D}}} (S_{τ_{D}} (t) B_{τ_{D}}^{n - 1}, B_{τ_{D}}^{n}) ⩽ L_{n} e^{- ω_{n} t}, \forall t ⩾ 0, \end{array}$ for some $L_{n} ⩾ 0$ and some $ω_{n} > 0$ .
Proof.
We need to show that the solution map is the sum of a term exponentially decreasing in the space $Ψ_{0}^{τ_{D}}$ , and a term uniformly bounded in $Ψ_{n}^{τ_{D}}$ . Adapting the proof of the Theorem 3.9 (in view of the proof of Lemma 4.2), it is immediate to see that the system (3.22)–(3.23) is exponentially stable in $Ψ_{0}^{τ_{D}}$ , whereas the solution w to (3.24)–(3.25) satisfies the uniform bound $\begin{array}{l} {‖ (w (t), w_{t}) ‖}_{Ψ_{n}^{τ_{D}}} ⩽ c, \forall t ⩾ 0 . \end{array}$ For some c depending on the radius $R_{n - 1}$ of $B_{τ_{D}}^{n - 1}$ . Redefining $R_{n}$ to be greater than or equal to the above constant c, we reach the desired conclusion. □

Till the end of the proof of Theorem 4.3, we agree to redefine inductively the radius $R_{n}$ so that Lemma 4.5 holds. Then, on account of the Lemma 3.7, Corollary 3.5, and Lemma 4.5, applying recursively Lemma 4.4, we get the result. □
Corollary 4.6.
Let ( 4.1 )–( 4.2 ) some hold for $1 ⩽ n ⩽ 4$ . Then the global attractor $A_{τ_{D}}$ is bounded in $Ψ_{n}^{τ_{D}}$ , with a bound independent of $τ_{D}$ .
Theorem 4.7.
Let ( 4.1 )–( 4.2 ) some hold for $1 ⩽ n ⩽ 4$ . Then, $\begin{array}{l} lim_{τ_{D} \to 0} [{dist}_{Ψ_{0}^{1}} (A_{τ_{D}}, {\tilde{A}}_{0})] = 0, \end{array}$ where $\begin{array}{l} {\tilde{A}}_{0} = {(u, ψ) : u \in A_{0}, ψ = - A (A u + f (u)) + g (u)} . \end{array}$

We point out that the Hausdorff semidistance in the above theorem is taken in $Ψ_{0}^{1}$ (and not just in $Ψ_{0}^{τ_{D}}$ ). Clearly, this is a stronger stability result.
5. Existence of exponential attractors

Provided that (4.1)–(4.2) hold for $n = 4$ , there is a robust family of exponential attractors ${M_{τ_{D}}}$ which is uniformly bounded in $Ψ_{n}^{τ_{D}}$ . Besides, the basin of attraction of each $M_{τ_{D}}$ coincides with the whole phase-space $Ψ_{0}^{τ_{D}}$ . In particular, $A_{τ_{D}} \subset M_{τ_{D}}$ .

We define the application $J$ by $\begin{array}{l} J : B_{0}^{4} & \to D (A^{- \frac{1}{2}}) \\ u & \mapsto J (u) = - A (A u + f (u)) + g (u) . \end{array}$ Introduce the lifting maps $L_{τ_{D}}$ : $B_{0}^{4} \to Ψ_{0}^{τ_{D}}$ as, $\begin{array}{l} L_{τ_{D}} u = \{\begin{matrix} (u, J u), & if τ_{D} > 0 \\ u, & if τ_{D} = 0 \end{matrix} \end{array}$

Remark 5.1.
Endowing $B_{0}^{4}$ with the metric topology of $D (A^{- \frac{1}{2}})$ , it is straightforward to check that $J$ is $\frac{1}{2}$ -Holder continuous from $B_{0}^{4}$ into $D (A^{- \frac{1}{2}})$ . Indeed, such a Hölder continuity is essential in order to apply the following Lemma 5.3 this is the reason we shall work in $B_{0}^{4}$ .
Theorem 5.2.
Let ( 4.1 )–( 4.2 ) hold for $n = 4$ . Then the semigroups $S_{τ_{D}} (t)$ possess compact positively invariant sets $M_{τ_{D}} \subset B_{0}^{4}$ “called exponential attractors” with fulfill the following conditions.
There exist $ω > 0$ and a positive increasing function $J$ such that, for any bounded set $B \subset Ψ_{0}^{τ_{D}}$ , there holds $\begin{array}{l} {dist}_{Ψ_{0}^{τ_{D}}} (S_{τ_{D}} (t) B, M_{τ_{D}}) ⩽ J (R) e^{- ω t}, \forall t ⩾ 0, \end{array}$ where $R = {sup}_{(u_{0}, u_{1}) \in B} ‖ (u_{0}, u_{1}) ‖_{Ψ_{0}^{τ_{D}}}$

The fractal dimension of $M_{τ_{D}}$ is uniformly bounded with respect to $τ_{D}$ .

There exist $ϵ \in (0, \frac{1}{8})$ and $C > 0$ such that $\begin{array}{l} {dist}_{Ψ_{0}^{τ_{D}}}^{sym} (M_{τ_{D}}, L_{τ_{D}} M_{0}) ⩽ C τ_{D}^{ϵ} . \end{array}$
The quantities ω, J, ϵ and C are independent of $τ_{D}$ . Here, $M_{τ_{D}}$ is the symmetric Hausdorff distance in $Ψ_{0}^{τ_{D}}$ .

The proof of the Theorem follows directly from the proof of the following Lemma. Lemma 5.3.
There exist $Δ_{i} ⩾ 0$ , $k \in (0, \frac{1}{2})$ and $t^{} ⩾ t_{n}$ (all independent of,* $τ_{D}$ such that the following conditions hold.
The map $S_{τ_{D}} = S_{τ_{D}} (t^{})$ satisfies, every* $z_{1} = (u_{0}^{1}, u_{1}^{1})$ , $z_{2} = (u_{0}^{2}, u_{1}^{2})$ in $B_{0}^{4}$ , $\begin{array}{l} S_{τ_{D}} (t) z_{1} - S_{τ_{D}} (t) z_{2} = L_{τ_{D}} (t) (z_{1}, z_{2}) + N_{τ_{D}} (t) (z_{1}, z_{2}), \end{array}$ where $\begin{array}{l} {‖ L_{τ_{D}} (t) (z_{1}, z_{2}) ‖}_{Ψ_{0}^{τ_{D}}} ⩽ k ‖ z_{1} - z_{2} ‖_{Ψ_{0}^{τ_{D}}} \\ {‖ N_{τ_{D}} (t) (z_{1}, z_{2}) ‖}_{Ψ_{1}^{τ_{D}}} ⩽ Δ_{1} ‖ z_{1} - z_{2} ‖_{Ψ_{0}^{τ_{D}}} . \end{array}$

The lifting map $L_{τ_{D}}$ fulfills $\begin{array}{l} {‖ S_{τ_{D}}^{k} (t) z - L_{τ_{D}} S_{0}^{k} (t) P_{τ_{D}} z ‖}_{Ψ_{0}^{τ_{D}}} ⩽ Δ_{2}^{k} \sqrt[8]{τ_{D}}, \forall z \in B_{0}^{4}, \forall k \in N, \end{array}$ where $P_{τ_{D}}$ : $B_{τ_{D}}^{4} \to B_{0}^{4}$ is the projection onto the first component when $τ_{D} > 0$ , and the identity map, otherwise.

For any $z \in B_{τ_{D}}^{4}$ , there holds $\begin{array}{l} {‖ S_{τ_{D}} (t) z - L_{τ_{D}} S_{0} (t) P_{τ_{D}} z ‖}_{Φ_{τ_{D}}^{0}} ⩽ Δ_{3} \sqrt[8]{τ_{D}}, \forall t \in [t^{}, 2 t^{}] . \end{array}$

The map $\begin{array}{l} (t, z) : \mapsto S_{τ_{D}} (t) z : [t^{}, 2 t^{}] \times B_{τ_{D}}^{4} \to B_{τ_{D}}^{4}, \end{array}$ is $(\frac{1}{2})$ -Hölder continuous. Besides, the map $z \mapsto S_{τ_{D}} (t) z$ is Lipschitz continuous on $B_{τ_{D}}^{4}$ , with a Lipschitz constant independent of $τ_{D}$ and $t \in [t^{}, 2 t^{}]$ .

Proof.
Throughout the proof of Lemma 5.3, from $c_{i}$ depend on the radius $R_{n}$ of the absorbing balls $B_{τ_{D}}^{4}$ .

Proof of L1. Let $z_{1} = (u_{0}^{1}, u_{1}^{1})$ , $z_{2} = (u_{0}^{2}, u_{1}^{2})$ in $B_{0}^{4}$ .

Then we write the difference $S_{τ_{D}} (t) z_{1} - S_{τ_{D}} (t) z_{2} = (\tilde{u}, {\tilde{u}}_{t}) = (\tilde{w} (t), {\tilde{w}}_{t}) + (\tilde{v}, {\tilde{v}}_{t})$ , where $(\tilde{w}, {\tilde{w}}_{t})$ and $(\tilde{v}, {\tilde{v}}_{t})$ are the solutions to the following problems $\begin{array}{l} (5.1) & τ_{D} {\tilde{v}}_{t t} + {\tilde{v}}_{t} + A^{2} \tilde{v} + A (f (\tilde{u}) - f (\tilde{w})) + \tilde{v} = g (\tilde{u}) - g (\tilde{w}) \\ (5.2) & \tilde{v} (0) = {\tilde{u}}_{0}, {\tilde{v}}_{t} (0) = {\tilde{u}}_{1} \end{array}$ and $\begin{array}{l} (5.3) & τ_{D} {\tilde{w}}_{t t} + {\tilde{w}}_{t} + A^{2} \tilde{w} + A f (\tilde{w}) + \tilde{w} - \tilde{u} = g (\tilde{w}) \\ (5.4) & \tilde{w} (0) = 0, {\tilde{w}}_{t} (0) = 0 . \end{array}$ Multiplying (5.3) by ${\tilde{w}}_{t}$ , we obtain $\begin{matrix} (5.5) & \frac{d}{d t} Γ (t) + 2 ‖ {\tilde{w}}_{t} ‖^{2} + 2 \int_{0}^{ℓ} {\tilde{w}}_{t} g^{'} (\tilde{w}) \tilde{w} d x = 2 (\tilde{u}, {\tilde{w}}_{t}) + \int_{0}^{ℓ} {\tilde{w}}_{t} f^{″} (\tilde{w}) | \tilde{w} |^{2} d x, \end{matrix}$ where $\begin{matrix} (5.6) & Γ (t) = τ_{D} ‖ {\tilde{w}}_{t} ‖^{2} + ‖ A \tilde{w} ‖^{2} + ‖ \tilde{w} ‖^{2} + \int_{0}^{ℓ} {| A^{\frac{1}{2}} \tilde{w} |}^{2} f^{'} (\tilde{w}) d x - \int_{0}^{ℓ} | \tilde{w} | g (\tilde{w}) d x . \end{matrix}$ Thanks to Young’s and Poincaré inequalities, we deduce $\begin{array}{l} \int_{0}^{ℓ} {| A^{\frac{1}{2}} \tilde{w} |}^{2} f^{'} (\tilde{w}) d x & ⩽ {‖ f^{'} (\tilde{w}) ‖}_{L^{\infty} (Ω)} {‖ A^{\frac{1}{2}} \tilde{w} ‖}^{2} \\ ⩽ c ‖ A \tilde{w} ‖^{2} + c^{'} \end{array}$ and $\begin{array}{l} \int_{0}^{ℓ} | \tilde{w} | g (\tilde{w}) d x & ⩽ ‖ g (\tilde{w}) ‖ ‖ \tilde{w} ‖ \\ ⩽ c ‖ \tilde{w} ‖_{L^{2 k} (Ω)}^{2 k} + c . \end{array}$ Combining the above estimates, we can rewrite (5.6) as follows $\begin{matrix} c_{1} (‖ A \tilde{w} ‖^{2} + τ_{D} ‖ {\tilde{w}}_{t} ‖^{2} + ‖ \tilde{w} ‖_{L^{2 k} (Ω)}^{2 k}) ⩽ Γ (t) ⩽ c_{2} (‖ A \tilde{w} ‖^{2} + τ_{D} ‖ {\tilde{w}}_{t} ‖^{2} + ‖ \tilde{w} ‖_{L^{2 k} (Ω)}^{2 k}) + c^{'} . \end{matrix}$ Thanks to Theorem 3.8 and Hölder’s inequality, we obtain $\begin{array}{l} | (\tilde{u}, {\tilde{w}}_{t}) | & ⩽ ‖ \tilde{u} ‖ ‖ {\tilde{w}}_{t} ‖ \\ ⩽ \frac{1}{2} ‖ {\tilde{w}}_{t} ‖^{2} + c^{'} ‖ \tilde{u} ‖^{2} . \end{array}$ Combinig the above estimates in (5.5), we have $\begin{matrix} (5.7) & \frac{d}{d t} Γ (t) + ‖ \tilde{w} ‖_{L^{2 k - 2} (Ω)}^{2 k - 2} ⩽ c Γ (t) + c^{'} ‖ \tilde{u} ‖^{2} . \end{matrix}$ Applying Gronwall’s Lemma to 5.7 and thanks to Lemma 3.7, we conclude $\begin{matrix} (5.8) & {‖ (\tilde{w}, {\tilde{w}}_{t}) ‖}_{Ψ_{1}^{τ_{D}}}^{2} ⩽ e^{k t} {‖ ({\tilde{u}}_{0}, {\tilde{u}}_{1}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} . \end{matrix}$ Multiplying (5.1) by $A^{- 1} (α \tilde{v} + {\tilde{v}}_{t})$ and thanks to Theorem (3.9), we obtain $\begin{matrix} (5.9) & {‖ (\tilde{v}, {\tilde{v}}_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} ⩽ e^{- k t} {‖ ({\tilde{u}}_{0}, {\tilde{u}}_{1}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} . \end{matrix}$ Taking $t^{} ⩾ t_{n}$ large enough, we deduce $\begin{array}{l} N_{τ_{D}} (z_{1}, z_{2}) = (\tilde{w} (t^{}), {\tilde{w}}_{t} (t^{})) \end{array}$ and $\begin{array}{l} L_{τ_{D}} (z_{1}, z_{2}) = (\tilde{v} (t^{}), {\tilde{v}}_{t} (t^{})), \end{array}$ fulfill (L1).

Proof of (L2) and (L3). Both assertions follow by the same form $\begin{matrix} (5.10) & {‖ S_{τ_{D}} (t) z - L_{τ_{D}} S_{0} (t) P_{τ_{D}} z ‖}_{Ψ_{0}^{τ_{D}}} ⩽ c e^{c t} \sqrt[4]{τ_{D}} + c e^{\frac{- t}{τ_{D}}}, \forall t ⩾ 0, z \in B_{τ_{D}}^{4} . \end{matrix}$ The proof of estimate, we need the following lemma Lemma 5.4.
For* $τ_{D} = 0$ , we have the following estimate $\begin{matrix} sup_{u_{0} \in B_{0}^{4}} \int_{0}^{\infty} {‖ u_{t t}^{0} (y) ‖}_{- 1}^{2} d y < \infty, \end{matrix}$ where $u^{0} (t) = S_{0} (t) u_{0}^{0}$ .
Proof.
Since $u_{0}$ is bounded in $D (A^{\frac{5}{2}})$ , then $J (u_{0})$ is bounded in $D (A^{\frac{1}{2}})$ . Consider the problem obtained by differentiating with respect to time (3.1) for $τ_{D} = 0$ , that is, $\begin{array}{l} (5.11) & ψ_{t} + A (A ψ + f^{'} (u^{0}) ψ) = ψ g^{'} (u^{0}) \end{array}$ and $\begin{array}{l} (5.12) & ψ (0) = J (u_{0}), \end{array}$ where $ψ = u_{t}^{0}$ .

It is a standard matter to see that there exists a unique solution for the problem (5.11)–(5.12) in $C ([0, T], D (A^{\frac{1}{2}})$ , for any $T > 0$ .

We multiply (5.11) by $α A^{- 1} ψ + A^{- 1} ψ_{t}$ , we obtain $\begin{array}{l} \frac{d}{d t} ({‖ A^{\frac{1}{2}} ψ ‖}^{2} + α {‖ A^{- \frac{1}{2}} ψ ‖}^{2}) + 2 α {‖ A^{\frac{1}{2}} ψ ‖}^{2} + 2 {‖ A^{- \frac{1}{2}} ψ_{t} ‖}^{2} \\ = - 2 α (A^{\frac{1}{2}} (ψ f^{'} (u^{0}), A^{- \frac{1}{2}} ψ) - 2 (A^{\frac{1}{2}} (ψ f^{'} (u^{0}), A^{- \frac{1}{2}} ψ_{t}) \\ (5.13) & + 2 α (g^{'} (u^{0}) ψ, A^{- 1} ψ) + 2 (g^{'} (u^{0}) ψ, A^{- 1} ψ_{t}) \end{array}$ On account of the controls $\begin{array}{l} ‖ A^{\frac{1}{2}} (ψ f^{'} (u^{0})) ‖ ⩽ c ‖ A^{\frac{1}{2}} ψ ‖ \end{array}$ and $\begin{array}{l} ‖ (g^{'} (u^{0}) ψ ‖ & ⩽ c ‖ g^{'} (u^{0}) ‖ ‖ ψ ‖ \\ ⩽ c ‖ A^{\frac{1}{2}} ψ ‖ {‖ u^{0} ‖}_{L^{2 k - 2} (Ω)}^{k - 1} \\ ⩽ c^{'} ‖ A^{\frac{1}{2}} ψ ‖ . \end{array}$ Combining the above estimates, we can rewrite (5.13) as $\begin{matrix} (5.14) & \frac{d}{d t} ({‖ A^{\frac{1}{2}} ψ ‖}^{2} + {‖ A^{- \frac{1}{2}} ψ ‖}^{2}) + {‖ A^{- \frac{1}{2}} \frac{d ψ}{d t} ‖}^{2} ⩽ c ({‖ A^{\frac{1}{2}} ψ ‖}^{2} + {‖ A^{- \frac{1}{2}} ψ ‖}^{2}) . \end{matrix}$ Applying Gronwall’s lemma to (5.14), we obtain the estimate. □

Let $(u_{0}, u_{1}) \in B_{0}^{4}$ .

We pose $\begin{matrix} (u^{τ_{D}} (t), u_{t}^{τ_{D}} (t)) = S_{τ_{D}} (t) (u_{0}, u_{1}) and (u^{0} (t), u_{t}^{0} (t)) = L_{τ_{D}} S_{0} (t) P_{τ_{D}} (u_{0}, u_{1}), \end{matrix}$ and $\begin{matrix} u = u^{τ_{D}} - u^{0} . \end{matrix}$ We have the following problem $\begin{array}{l} (5.15) & τ_{D} u_{t t} + u_{t} + A^{2} u + A (f (u^{τ_{D}}) - f (u^{0}) = - τ_{D} u_{t t}^{0} + g (u^{τ_{D}}) - g (u^{0}) \\ (5.16) & u (0) = 0, u_{t} (0) = u_{1} - J (u_{0}) . \end{array}$ Multiplying (5.11) by $A^{- 1} u_{t}$ , we get $\begin{array}{l} \frac{d}{d t} [τ_{D} ‖ u_{t} ‖_{- 1}^{2}] + 2 ‖ u_{t} ‖_{- 1}^{2} = & - 2 (A^{\frac{3}{2}} u, A^{- \frac{1}{2}} u_{t}) - 2 τ_{D} (u_{t t}^{0}, A^{- 1} u_{t}) \\ (5.17) & + 2 (g (u^{τ_{D}}) - g (u^{0}), A^{- 1} u_{t}) - 2 (f (u^{τ_{D}}) - f (u^{0}), u_{t}) . \end{array}$ Thanks to Lemma 4.2, Young’s and Poincaré inequalities, we deduce $\begin{array}{l} - 2 τ_{D} | (u_{t t}^{0}, A^{- 1} u_{t}) | & ⩽ τ_{D} ‖ A^{- \frac{1}{2}} u_{t t}^{0} ‖ ‖ A^{- \frac{1}{2}} u_{t} ‖ \\ ⩽ c^{'} τ_{D} {‖ u_{t t}^{0} ‖}_{- 1}^{2} + \frac{1}{4} ‖ u_{t} ‖_{- 1}^{2} \end{array}$ and $\begin{array}{l} - 2 | (A^{\frac{3}{2}} u, A^{- \frac{1}{2}} u_{t}) | & ⩽ c ‖ A^{\frac{3}{2}} u ‖ ‖ A^{- \frac{1}{2}} u_{t} ‖ \\ ⩽ c^{'} + \frac{1}{4} ‖ u_{t} ‖_{- 1}^{2} . \end{array}$ Exploiting the Theorem 3.2 and Lemma 4.2, we conclude $\begin{array}{l} - 2 | (A^{\frac{1}{2}} (f (u^{τ_{D}}) - f (u^{0})), A^{- \frac{1}{2}} u_{t}) | ⩽ c ‖ A^{\frac{1}{2}} u ‖ ‖ A^{- \frac{1}{2}} u_{t} ‖ \\ ⩽ c^{″} + \frac{1}{4} ‖ u_{t} ‖_{- 1}^{2} \\ 2 | (g (u^{τ_{D}}) - g (u^{0}), A^{- 1} u_{t}) | ⩽ c ‖ A^{\frac{1}{2}} u ‖ ‖ A^{- \frac{1}{2}} u_{t} ‖ \\ ⩽ c^{‴} + \frac{1}{4} ‖ u_{t} ‖_{- 1}^{2} . \end{array}$ Combining the above estimates in (5.13), we deduce $\begin{matrix} (5.18) & \frac{d}{d t} (τ_{D} ‖ u_{t} ‖_{- 1}^{2}) + \frac{1}{τ_{D}} (τ_{D} ‖ u_{t} ‖_{- 1}^{2}) ⩽ c_{1} + c^{'} τ_{D} {‖ u_{t t}^{0} ‖}_{- 1}^{2} . \end{matrix}$ Thanks to Gronwall’s lemma, Lemma 5.4, we obtain $\begin{matrix} (5.19) & τ_{D} ‖ u_{t} ‖_{- 1}^{2} ⩽ e^{- \frac{t}{τ_{D}}} + c^{'} τ_{D} . \end{matrix}$ We will rewrite (5.17), we have $\begin{array}{l} \frac{d}{d t} {‖ (u, u_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} + 2 ‖ u_{t} ‖_{- 1}^{2} \\ (5.20) & = - 2 τ_{D} (u_{t t}^{0}, A^{- 1} u_{t}) + 2 (g (u^{τ_{D}}) - g (u^{0}), A^{- 1} u_{t}) - 2 (f (u^{τ_{D}}) - f (u^{0}), u_{t}) . \end{array}$ Combinig the above estimates in (5.20),we find $\begin{matrix} (5.21) & \frac{d}{d t} {‖ (u, u_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} ⩽ c {‖ (u, u_{t}) ‖}_{Ψ_{0}^{τ_{D}}}^{2} + c^{'} τ_{D} {‖ u_{t t}^{0} ‖}_{- 1}^{2} . \end{matrix}$ Applying Gronwall’s lemma to (5.21) on $(\sqrt{τ_{D}}, t + \sqrt{τ_{D}})$ , along with (5.19) and Lemma 5.4, we obtain $\begin{matrix} {‖ A^{\frac{1}{2}} u (t + \sqrt{τ_{D}}) ‖}^{2} ⩽ e^{c t} ({‖ A^{\frac{1}{2}} u (\sqrt{τ_{D}}) ‖}^{2} + c e^{- \frac{1}{\sqrt{τ_{D}}}} + c^{'} τ_{D}) . \end{matrix}$ Thus, we learn that $\begin{matrix} {‖ A^{\frac{1}{2}} u (t) ‖}^{2} ⩽ e^{c t} ({‖ A^{\frac{1}{2}} u (\sqrt{τ_{D}}) ‖}^{2} + c^{'} τ_{D}) . \end{matrix}$ Thanks to Lemma 4.2 and interpolation inequality, we deduce $\begin{array}{l} {‖ A^{\frac{1}{2}} u (t) ‖}^{2} & ⩽ ‖ A^{\frac{3}{2}} u (t) ‖ ‖ A^{- \frac{1}{2}} u (t) ‖ \\ ⩽ c \int_{0}^{t} {‖ u_{t} (y) ‖}_{- 1} d y \\ ⩽ c \sqrt{t} . \end{array}$ Finally, collecting the two above inequalities, we conclude that $\begin{array}{l} (5.22) & ‖ A^{\frac{1}{2}} u (t) ‖ ⩽ c e^{c t} \sqrt[4]{τ_{D}} . \end{array}$ Combining (5.22), (5.19) we obtain the estimation result.

Hence, the proof of $(L 2)$ and $(L 3)$ .

Proof of the (L4). For any $t^{'}, t \in [t^{}, 2 t^{}]$ and for any $z_{1}, z_{2} \in B_{τ_{D}}^{4}$ , in view of Lemma 3.7, there holds $\begin{array}{l} {‖ S_{τ_{D}} (t) z_{1} - S_{τ_{D}} (t^{'}) z_{2} ‖}_{Ψ_{0}^{τ_{D}}} & ⩽ {‖ S_{τ_{D}} (t) z_{1} - S_{τ_{D}} (t^{'}) z_{1} ‖}_{Ψ_{0}^{τ_{D}}} + {‖ S_{τ_{D}} (t^{'}) z_{1} - S_{τ_{D}} (t^{'}) z_{2} ‖}_{Ψ_{0}^{τ_{D}}} \\ ⩽ {‖ S_{τ_{D}} (t) z_{1} - S_{τ_{D}} (t^{'}) z_{1} ‖}_{Ψ_{0}^{τ_{D}}} + e^{2 K t^{}} ‖ z_{1} - z_{2} ‖_{Ψ_{0}^{τ_{D}}} . \end{array}$ Setting $t = t^{'}$ , we get at once the second assertion of $(L 4)$ . In the proof of the first one, we will show that $\begin{matrix} (5.23) & sup_{z \in B_{τ_{D}}^{4}} \int_{t^{}}^{2 t^{}} {‖ (u_{t} (y), u_{t t} (y)) ‖}_{Ψ_{0}^{τ_{D}}}^{2} d y ⩽ c, \end{matrix}$ where, as usual, $u (t)$ denotes the first component of $S_{τ_{D}} (t) z$ .

We multiply (3.1) by $A u_{t}$ , we have $\begin{matrix} (5.24) & \frac{d}{d t} (τ_{D} {‖ A^{\frac{1}{2}} u_{t} ‖}^{2} + {‖ A^{\frac{3}{2}} u ‖}^{2}) + 2 {‖ A^{\frac{1}{2}} u_{t} ‖}^{2} = - 2 (A^{\frac{3}{2}} f (u), A^{\frac{1}{2}} u_{t}) + 2 (A^{\frac{1}{2}} g (u), A^{\frac{1}{2}} u_{t}) . \end{matrix}$ Thanks to Lemma 4.2, we have $\begin{array}{l} \frac{d}{d t} (τ_{D} {‖ A^{\frac{1}{2}} u_{t} ‖}^{2} + {‖ A^{\frac{3}{2}} u ‖}^{2}) + {‖ A^{\frac{1}{2}} u_{t} ‖}^{2} ⩽ c . \end{array}$ Hence, integrating on $(t^{}, 2 t^{})$ , we find $\begin{array}{l} (5.25) & \int_{t^{}}^{2 t^{}} {‖ A^{\frac{1}{2}} u_{t} (y) ‖}^{2} d y ⩽ c . \end{array}$ Multiplying (3.1) by $A^{- 1} u_{t t}$ , we obtain $\begin{array}{l} \frac{d}{d t} ({‖ A^{- \frac{1}{2}} u_{t} ‖}^{2} + 2 (A^{\frac{3}{2}} u, A^{- \frac{1}{2}} u_{t}) + 2 (A^{\frac{1}{2}} f (u), A^{- \frac{1}{2}} u_{t})) + 2 τ_{D} {‖ A^{- \frac{1}{2}} u_{t t} ‖}^{2} \\ (5.26) & = 2 {‖ A^{\frac{1}{2}} u_{t} ‖}^{2} + 2 (f^{'} (u) u_{t}, u_{t}) + 2 (g (u), A^{- 1} u_{t t}) . \end{array}$ On account of the above estimates, an integrating over $(t^{}, 2 t^{*})$ yields the remaining part of (5.23). □
Remark 5.5.
Actually, the coefficient in the Hölder continuity does not need to be independent of $τ_{D}$ to deduce the uniform (with respect to $τ_{D}$ ) finite fractal dimensionality of the family of robust exponential attractors $M_{τ_{D}}$ . Recall that the fractal dimension of a (relatively) compact set $K$ in a metric space $X$ is defined by $\begin{array}{l} {dim}_{F} (K, X) = \underset{μ \to 0}{lim sup} \frac{H_{μ} (K, X)}{log (\frac{1}{μ})} \end{array}$ where $H_{μ} (K, X) = log N_{μ} (H_{μ} (K, X)$ denotes the so-called Kolmogorov μ-entropy of $K$ in $X$ , $N_{μ} (H_{μ} (K, X)$ being the minimal number of the μ-balls of $X$ to cover $K$ . So, if the constant in the Hölder continuity is independent of $τ_{D}$ , as it is the case here, we further deduce that the Kolmogorov μ-entropy of the family of robust exponential attractors is, for fixed $μ > 0$ , bounded from above independently of $τ_{D}$ .

6. Numerical results

For the numerical simulations of the model, we rewrite (3.1) in the following equivalent form: $\begin{array}{l} (H) \{\begin{matrix} τ_{D} u_{t t} + u_{t} - Δ μ = g (u), \\ μ = - Δ u + f (u), \end{matrix} \end{array}$ where $τ_{D}$ is the relaxation parameter.

Fig. 1.

Evolution of the solution after every 10 iterations. In this case we take $N = 70$ , $λ_{d} = 240$ , $λ_{g} = 560$ and $τ_{D} = 0$ i.e., the classical Cahn–Hilliard equation with the prolifertion term. The time step is $Δ t = {5.10}^{- 5}$ .

Fig. 2.

Evolution of the solution after every 230 iterations. In this case we take $N = 1650$ , $λ_{d} = 240$ , $λ_{g} = 560$ and $τ_{D} = 0.1$ i.e., the hyperbolic relaxation of the Cahn–Hilliard equation with the proliferation term. The time step is $Δ t = {5.10}^{- 5}$ .

Fig. 3.

Evolution of the solution after every 50 iterations. In this case we take $N = 350$ , $λ_{d} = 240$ , $λ_{g} = 560$ and $τ_{D} = 0.01$ i.e., the hyperbolic relaxation of the Cahn–Hilliard equation with the proliferation term. The time step is $Δ t = {5.10}^{- 5}$ .

Fig. 4.

Evolution of $u_{\max}$ , $u_{\min}$ and $u_{mean}$ for 70 iterations. In this case we take $τ_{D} = 0.0$ and $λ_{d} = 240$ , $λ_{g} = 560$ i.e., the classical Cahn–Hilliard equation with the proliferation term. The time step is $Δ t = {5.10}^{- 5}$ .

We consider a $P_{1}$ finite element approach and a semi-implicit Euler time discretization. Let $T_{h}$ be the triangulation of $\overline{Ω}$ and $V_{h} = {v_{h} \in C^{0} (\overline{Ω}); v_{h} = 0 o n \partial \overline{Ω} v_{h} |_{K} \in P_{1}, \forall K \in T_{h}}$ the finite element space. We denote by $Δ t > 0$ the time step. The variational formulation of the semi-discretized model is as follows.

Let $u^{0}, u^{1} \in V_{h}$ be given and for $n = 1, 2, \dots$ , find $u^{n + 1}, μ^{n + 1} \in H_{0}^{1} (Ω) = V$ , such that $\begin{array}{l} (6.1) & (τ_{D} \frac{u^{n + 1} - 2 u^{n} + u^{n - 1}}{Δ t^{2}}, χ) + (\frac{u^{n + 1} - u^{n}}{Δ t}, χ) + (\nabla μ^{n + 1}, \nabla χ) = (g (u^{n}), χ), \\ (6.2) & (\nabla u^{n + 1}, \nabla ψ) - (μ^{n + 1}, ψ) = - (f (u^{n}), ψ), \end{array}$ for all $χ, ψ \in V_{h}$ (note that $μ^{0}$ and $μ^{1}$ are not needed). Remark 6.1.

In the case $g = 0$ , the stability result of the scheme (6.1)–(6.2) was obtained in [15,17].

Fig. 5.

Evolution of $u_{\max}$ , $u_{\min}$ and $u_{mean}$ for 10000 iterations. In this case we take $λ_{d} = 240$ , $λ_{g} = 560$ and $τ_{D} = 0.1$ i.e., the hyperbolic relaxation of the Cahn–Hilliard equation with the proliferation term. The time step is $Δ t = {5.10}^{- 5}$ .

Fig. 6.

Evolution of $u_{\max}$ , $u_{\min}$ and $u_{mean}$ for 2000 iterations. In this case we take $λ_{d} = 240$ , $λ_{g} = 560$ and $τ_{D} = 0.01$ i.e., the hyperbolic relaxation of the Cahn–Hilliard equation with the proliferation term. The time step is $Δ t = {5.10}^{- 5}$ .

6.1. Numerical simulations in one space dimension

The numerical simulations are performed with software Scilab (http://www.scilab.org/). In the numerical results presented below, the domain is $Ω = [0, 1]$ . The mesh size is equal to $h = \frac{1}{500}$ and the time step to $δ t = 0.00005$ . We choose $λ_{d} = 240$ , $M = 500$ , $ϵ = 0.1$ , $f (s) = \frac{1}{ϵ^{2}} (s^{3} - s)$ and $g (s) = - λ_{d} (s + 1) + λ_{g} {(s + 1)}^{2} {(s - 1)}^{2}$ where $λ_{g}$ and $λ_{d}$ are the growth and death coefficients, respectively, of the tumor cells.

In Figs 1–3, we present the evolution of the solution for different iterations under the influence of the parameters $τ_{D}$ , and $λ_{g}$ .

In Figs 4–6, we show the influence of the parameter’s $τ_{D}$ , $λ_{g}$ and $λ_{d}$ on the evolution of the decomposition of phases. We take the initial condition random on a coarse mesh. In the three cases, the solution u of the (2.1)–(2.3) evolves from $u_{0}$ to a steady state. The parameter of relaxation $τ_{D}$ corresponds to a typical damping time of the oscillations. We can see in Fig. 5, we put 5 times more time than in Fig. 6 for the damping.

Footnotes

Acknowledgements

The author is thankful to Morgan PIERRE and Alain MIRANVILLE for many helpful discussions.

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