Robust exponential attractors for a parabolic–hyperbolic phase-field system

Abstract

In this paper, we construct a robust family of exponential attractors for a parabolic–hyperbolic phase-field system (PHPFS), which describes phase separation in material sciences, e.g., melting and solidification. A consequence of this is the existence of finite fractal dimensional global attractors which are both upper and lower semicontinuous at the parameter $\epsilon=0$. Hence we establish the convergence of the dynamics of PHPFS to those of the well known Cagilnap phase-field system.

1 Introduction

Exponential attractors are compact and positively invariant sets with finite fractal dimension which attract all the trajectories starting from bounded sets at a uniform exponential rate (see [5–7, 14]). The existence of exponential attractors guarantees the existence of a finite fractal dimensional global attractor. Readers may see [4, 8, 13] and references therein for more on the dimension of a global attractor. Thus a finite-dimensional reduction principle can be applied to reduce the infinite-dimensional dynamical system under consideration to a finite-dimensional system of ODEs. The sensitivity of exponential attractors under small perturbations is the main focus in this work. One may see [15] for some recent developments in the construction of exponential attractors.

The phase-field system is a system of equations which couples the temperature u and order parameter ϕ also known as “phase-field”. It describes phase separations in materials occupying a domain $\Omega\subset\mathbb{R}^{d}$.

We consider the following parabolic–hyperbolic phase-field system (PHPFS):

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi+ \phi+ g(\phi)-u=0,\\ u_{t}+\phi_{t}-\Delta u =0,\\ \partial_{n}\phi|_{\partial\Omega}=u|_{\partial\Omega}=0,\\ \phi(0)=\phi_{0}, \phi_{t}(0)=\phi_{1}, u(0)=u_{0}, \end{cases} $$

(1.1)

in a bounded domain $\Omega\subset\mathbb{R}^{d}$, $d=1,2,3$ with smooth boundary ∂Ω, where $\epsilon\in(0,1]$ is a small parameter. Denote the function $G(s)=\int_{0}^{s}g(\varsigma)\,d\varsigma$ and assume that g satisfies $g\in\mathcal{ C}^{2}(\mathbb{R})$ and the following conditions hold (cf., e.g., [1, 3]):

$$\begin{aligned} & G(s)\ge-C_{1},\qquad C_{1}\geq0, \quad \forall s\in\mathbb{ R}, \end{aligned}$$

(1.2)

$$\begin{aligned} &\forall\gamma\in\mathbb{ R}, \exists C_{2}(\gamma)> 0,C_{3}(\gamma)\ge 0 \mbox{ such that} \\ & (s-\gamma)g(s)-C_{2}G(s)\ge-C_{3}, \quad\forall s\in \mathbb{ R} \end{aligned}$$

(1.3)

$$\begin{aligned} &(\mbox{where } C_{2},C_{3} \mbox{ are bounded when }\gamma \mbox{ is bounded}), \\ &g'(s)\geq-C_{4},\qquad C_{4} \geq0, \quad\forall s\in\mathbb{ R}, \end{aligned}$$

(1.4)

$$\begin{aligned} & \bigl\vert g''(s) \bigr\vert \leq C_{5} \bigl( \vert s \vert ^{p}+1 \bigr), \qquad C_{5}> 0, \quad\forall s\in\mathbb{ R}, \end{aligned}$$

(1.5)

with $p\geq0$ when $d=1,2$ and $p\in[0,1]$ when $d=3$. We note that in space dimension one, no growth assumption on g is needed.

We remark that our results also hold when ϕ is subject to a boundary condition of periodic type

$$ \textstyle\begin{cases} u|_{x_{i}=0}=u|_{x_{i}=L_{i}}, \qquad u_{x_{i}}|_{x_{i}=0}=u_{x_{i}}|_{x_{i}=L_{i}}, \quad i=1,\dots,d,\\ \phi|_{x_{i}=0}=\phi|_{x_{i}=L_{i}}, \quad i=1,\dots,d,\\ \quad\mbox{for }\phi\mbox{ and the derivatives of }\phi\mbox{ of order}\le3, \end{cases} $$

(1.6)

if $\Omega=\prod_{i=1}^{d}(0,L_{i})$.

We shall construct a robust family of exponential attractors which are both upper and lower semicontinuous at $\epsilon=0$ with respect to a norm independent of ϵ.

Grasselli and Pata [10] showed a well-posedness result and the existence of the global attractor for the system ($\epsilon>0$)

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi+\phi^{3}=\gamma(\phi)+\lambda '(\phi)u,\\ u_{t}+\lambda'(\phi)\phi_{t}-\Delta u=f. \end{cases} $$

Grasselli and Pata [11] considered the system ($\epsilon>0$)

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi+\phi-\lambda'(\phi)u+h(\phi )=\xi,\\ u_{t}+\lambda'(\phi)\phi_{t}-\Delta u=0 \end{cases} $$

(1.7)

in 3D, subject to mixed boundary conditions, Neumann on ϕ and Dirichlet on u. They proved a well-posedness result, the existence of the global attractor and its upper semicontinuity at $\epsilon=0$, and constructed exponential attractors with respect to a norm depending on ϵ. Also, Grasselli et al. [9] gave a well-posedness result and constructed a robust family of exponential attractors ${\mathbb {E}}_{\epsilon}$ for the system

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi-\lambda'(\phi)u+\chi(\phi )=\xi,\\ u_{t}+\lambda'(\phi)\phi_{t}-\Delta u=0 \end{cases} $$

(1.8)

in 3D, subject to Dirichlet boundary conditions on both ϕ and u, where $\chi(\phi)$ is singular at $\phi=\pm1$, e.g., $\ln (\frac{1+\phi}{1-\phi} )$, $\phi\in (0,1)$. More precisely, they showed that there exist $c>0$ and $\varpi\in(0,1)$, both independent of ϵ, such that

$$\begin{aligned} \operatorname{dist}^{\mathrm{sym}}_{K,\epsilon}({\mathbb {E}}_{\epsilon}, { \mathbb {E}}_{0})\leq c\epsilon^{\varpi},\quad\forall\epsilon \in[0,1], \end{aligned}$$

in the norm $\|(\phi,\phi_{t},u)\|^{2}_{K,\epsilon}=\|\Delta\phi\| _{L^{2}(\Omega)}^{2}+\epsilon\|\nabla\phi_{t}\|_{L^{2}(\Omega)}^{2}+\|\phi _{t}\|_{L^{2}(\Omega)}^{2} +\|\Delta u\|_{L^{2}(\Omega)}^{2}$, which clearly depends on ϵ.

Finally, we would also like to mention the papers [12, 16, 17] where the convergence to equilibrium of solutions for a parabolic–hyperbolic phase-field model were proven.

This work is organized as follows. In Sect. 1, we give a brief introduction. In Sect. 2, we give some a priori estimates. In Sect. 3, we construct exponential attractors for the system (1.1). Finally, in Sect. 4, we construct a robust family of exponential attractors which are both upper and lower semicontinuous at $\epsilon =0$ for the system (1.1).

We define the Hilbert space ${\mathcal {H}}_{r,\epsilon}=H^{r}\times H^{r-1}\times H^{r-1}_{0}$, $r\geq1$, endowed with the norm

$$\bigl\Vert (\varphi,\psi,v) \bigr\Vert _{{\mathcal {H}}_{r,\epsilon}}=\bigl( \Vert \varphi \Vert _{r}^{2}+\epsilon \Vert \psi \Vert _{r-1}^{2}+ \Vert v \Vert _{r-1}^{2} \bigr)^{1/2}, $$

where we understand that $H^{0}_{0}=H^{0}(\Omega)=L^{2}(\Omega)$. Hence, we denote ${\mathcal {H}}_{1,0}=H^{1}(\Omega)\times L^{2}(\Omega)$, endowed with the norm $\|(\cdot,\cdot)\|_{{\mathcal {H}}_{1,0}}=(\|\cdot\|_{1}^{2}+\|\cdot \|^{2})^{1/2}$.

2 A priori estimates

We multiply (1.1)₁ by $\phi_{t}$ and (1.1)₂ by u, then integrate over Ω. Adding the resulting equations, we obtain

$$\begin{aligned} \frac{d}{dt}E_{1}(t)+ 2 \Vert \phi_{t} \Vert ^{2}+2 \Vert \nabla u \Vert ^{2}=0 \end{aligned}$$

(2.1)

where

$$E_{1}(t)= \Vert \nabla\phi \Vert ^{2}+ \Vert \phi \Vert ^{2}+\epsilon \Vert \phi_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+2 \int_{\Omega}G(\phi)\,dx. $$

From (1.2), (1.3) and (1.5), we deduce that

$$\int_{\Omega}G(\phi)\,dx\geq-C_{1} \vert \Omega \vert \quad\mbox{and}\quad \int _{\Omega}G(\phi)\,dx\leq c\bigl( \Vert \phi \Vert _{1}^{p+3}+1\bigr). $$

Hence,

$$\begin{aligned} \bigl\Vert (\phi,\phi_{t},u) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}-\alpha_{1}\leq E_{1}(t)\leq \alpha_{2}\bigl( \Vert \phi \Vert _{1}^{p+3}+ \epsilon \Vert \phi_{t} \Vert ^{2}+ \Vert u \Vert ^{2}+1\bigr), \end{aligned}$$

(2.2)

for some $\alpha_{1},\alpha_{2}>0$ independent of ϵ. Thus integrating (2.1) over $(0,t)$ and accounting for (2.2), we obtain that

$$\int_{0}^{t}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(s) \bigr\Vert ^{2} \bigr)\,ds\leq E_{1}(0)+\alpha _{1},\quad\forall t\geq0. $$

Hence by (2.2) again, we get

$$\begin{aligned} \int_{0}^{\infty}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(s) \bigr\Vert ^{2}\bigr)\,ds\leq c\bigl( \Vert \phi_{0} \Vert _{1}^{p+3}+\epsilon \Vert \phi_{1} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+1\bigr). \end{aligned}$$

(2.3)

Let $(\phi^{1},u^{1})$ and $(\phi^{2},u^{2})$ be two solutions of (1.1). Set $\phi=\phi^{1}-\phi^{2}$, $\phi_{t}=\phi_{t}^{1}-\phi_{t}^{2}$ and $u=u^{1}-u^{2}$, then $\phi(0)=0$, $\phi_{t}(0)=0$ and $u(0)=0$. The pair $(\phi,\phi_{t},u)$ is a solution to the problem

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi+\phi+g(\phi^{1})-g(\phi ^{2})-u=0,\\ u_{t}+\phi_{t}-\Delta u=0,\\ \phi(0)=\phi_{t}(0)=u(0)=0. \end{cases} $$

(2.4)

We multiply (2.4)₁ and (2.4)₂ by $\phi_{t}$ and u, respectively, integrate over Ω, then add the resulting equations to get

$$ \frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla\phi \Vert ^{2}+ \Vert \phi \Vert ^{2}+\epsilon \Vert \phi_{t} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr)+ \Vert \phi_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2}=-\bigl(g\bigl(\phi ^{1}\bigr)-g\bigl( \phi^{2}\bigr),\phi_{t}\bigr). $$

By Hölder’s inequality and (1.5), we have

$$\begin{aligned} \bigl\vert \bigl(g\bigl(\phi^{1}\bigr)-g\bigl(\phi^{2} \bigr),\phi_{t}\bigr) \bigr\vert & \leq c\bigl( \bigl\Vert \phi^{1} \bigr\Vert _{L^{3p+3}(\Omega)}^{p+1}+ \bigl\Vert \phi^{2} \bigr\Vert _{L^{3p+3}(\Omega)}^{p+1}+1 \bigr) \Vert \phi \Vert _{L^{6}(\Omega)} \Vert \phi_{t} \Vert \\ &\leq c\bigl( \bigl\Vert \phi^{1} \bigr\Vert _{1}^{p+1}+ \bigl\Vert \phi^{2} \bigr\Vert _{1}^{p+1}+1 \bigr) \Vert \phi \Vert _{1} \Vert \phi_{t} \Vert . \end{aligned}$$

Therefore, by Young’s inequality, we obtain

$$ \frac{d}{dt} \bigl( \Vert \nabla\phi \Vert ^{2}+ \Vert \phi \Vert ^{2}+\epsilon \Vert \phi_{t} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr) \leq{\widetilde{M}}(t) \Vert \phi \Vert _{1}^{2}, $$

(2.5)

where

$${\widetilde{M}}(t)= \textstyle\begin{cases} c\sup_{\theta\in[0,1]} \Vert g'(\theta\phi_{1}+(1-\theta)\phi_{2}) \Vert ^{2}_{L^{\infty}(\Omega)},&\mbox{if }d=1,\\ c( \Vert \phi^{1} \Vert _{1}^{2p+2}+ \Vert \phi^{2} \Vert _{1}^{2p+2}+1),&\mbox{if }d=2,3. \end{cases} $$

Noting that $t\mapsto{\widetilde{M}}(t)$ is $L^{1}(0,T)$, and integrating (2.5) over $(0,t)$, we deduce that

$$ \bigl\Vert \bigl(\phi(t),\phi_{t}(t),u(t)\bigr) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}} \leq e^{\int_{0}^{t}{\widetilde{M}}(s)\,ds} \bigl\Vert \bigl( \phi(0),\phi_{t}(0),u(0)\bigr) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}, \quad\forall t\geq0. $$

(2.6)

We state a well-posedness result, which is proved in [11, Theorem 3.4].

Theorem 2.1

We assume that (1.2)–(1.5) hold. If $(\phi_{0},\phi_{1},u_{0})\in{\mathcal {H}}_{1,\epsilon}$, then (1.1) possesses a unique solution $(\phi,u)$ such that

$$\begin{aligned} (\phi,\phi_{t},u)\in{\mathcal {C}}\bigl([0,T]; {\mathcal {H}}_{1,\epsilon}\bigr) \end{aligned}$$

for any $T>0$. Moreover, if $(\phi_{0},\phi_{1},u_{0})\in{\mathcal {H}}_{2,\epsilon}$, then $(\phi,\phi_{t}, u) \in{\mathcal {C}}([0,T];{\mathcal {H}}_{2,\epsilon})$.

On account of Theorem 2.1 we can define the semigroup

$$S_{\epsilon}(t):{\mathcal {H}}_{1,\epsilon}\rightarrow{\mathcal {H}}_{1,\epsilon},\qquad(\phi_{0},\phi_{1},u_{0}) \mapsto\bigl(\phi(t),\phi _{t}(t),u(t)\bigr),\quad\forall t\geq0, $$

where $(\phi(t),\phi_{t}(t),u(t))$ is the solution to problem (1.1) at time t. The semigroup $S_{\epsilon}(t)$ is strongly continuous (cf. (2.6)).

It is also known from [11] that the semigroup $S_{\epsilon}(t):{\mathcal {H}}_{j,\epsilon}\rightarrow{\mathcal {H}}_{j,\epsilon}$ has bounded absorbing sets ${\mathcal {B}}_{j}$ in ${\mathcal {H}}_{j,\epsilon}$ of the form

$$\begin{aligned} &{\mathcal {B}}_{j}=\bigl\{ (\varphi,\psi,v)\in{\mathcal {H}}_{j,\epsilon}, \bigl\Vert (\varphi,\psi,v) \bigr\Vert _{{\mathcal {H}}_{j,\epsilon}}\leq r_{j}\bigr\} ,\quad j=1,2, \end{aligned}$$

where $r_{j}>0$ is independent of ϵ. In fact, they are exponentially attracting sets.

3 Exponential attractors

Now we state sufficient conditions which guarantee the existence of robust exponential attractors, which are continuous with respect to ϵ (cf. [2, Theorem 5.1]; also [1, 7, 15]).

Theorem 3.1

([2])

Let $E^{1}$, $E^{2}$, $V^{1}$, $V^{2}$, $W^{1}$, $W^{2}$ be Banach spaces such that $W^{i}\Subset V^{i}\Subset E^{i}$, $i=1,2$. Set $E_{\epsilon}=E^{1}\times E^{2}$, $V_{\epsilon}=V^{1}\times V^{2}$, $W_{\epsilon}=W^{1}\times W^{2}$ and endow them with the following norms:

$$\begin{aligned}& \bigl\Vert (p,q) \bigr\Vert _{E_{\epsilon}} =\bigl( \Vert p \Vert _{E^{1}}^{2} +\epsilon \Vert q \Vert _{E^{2}}^{2} \bigr)^{1/2}, \\& \bigl\Vert (p,q) \bigr\Vert _{V_{\epsilon}} =\bigl( \Vert p \Vert _{V^{1}}^{2} +\epsilon \Vert q \Vert _{V^{2}}^{2} \bigr)^{1/2}, \\& \bigl\Vert (p,q) \bigr\Vert _{W_{\epsilon}} =\bigl( \Vert p \Vert _{W^{1}}^{2} +\epsilon \Vert q \Vert _{W^{2}}^{2} \bigr)^{1/2}, \end{aligned}$$

respectively, where $\epsilon\in[0,1]$, with the convention that $E_{0}=E^{1}$, $V_{0}=V^{1}$, and $W_{0}=W^{1}$. Let $B_{\epsilon}(r)$ denote a closed ball in $W_{\epsilon}$ of radius $r>0$ and centered at zero. Consider a one-parameter family of strongly continuous semigroups $\{ S_{\epsilon}(t)\}_{\epsilon}$ acting on the phase-space $E_{\epsilon}$, for each $\epsilon\in[0,1]$. Then assume that there exist $\alpha,\beta,\gamma,\vartheta\in (0,1]$, $\kappa\in(0,\frac{1}{2})$, $\Upsilon_{j}\geq0$, and $\varrho>0$ (all independent of ϵ) such that, setting $B_{\epsilon}=B_{\epsilon}(\varrho)$, the following conditions hold:

• 1. There exists a map ${\mathscr {L}}: B_{0}\rightarrow V^{2}$ which is Hölder continuous of exponent α. Here $B_{0}$ is endowed with the metric topology of $E^{1}$.

• 2. There exists $t^{\star}>0$, independent of ϵ, such that

  $$ S_{\epsilon}(t) B_{\epsilon}\subset B_{\epsilon}, \quad\forall t\geq t^{\star}, $$

  and $B_{\epsilon}$ is uniformly bounded (with respect to ϵ) in the $E_{1}$-norm. Moreover, setting $S_{\epsilon}(t^{\star})=S_{\epsilon}$, the map $S_{\epsilon}$ satisfies, for every $z_{1},z_{2}\in B_{\epsilon}$,

  $$S_{\epsilon}z_{1}-S_{\epsilon}z_{2}=L_{\epsilon}(z_{1},z_{2})+K_{\epsilon}(z_{1},z_{2}), $$

  where

  $$\begin{aligned} & \Vert L_{\epsilon}z_{1}-L_{\epsilon}z_{2} \Vert _{E_{\epsilon}}\leq\kappa \Vert z_{1}-z_{2} \Vert _{E_{\epsilon}}, \\ & \Vert K_{\epsilon}z_{1}-K_{\epsilon}z_{2} \Vert _{V_{\epsilon}}\leq\Upsilon_{1} \Vert z_{1}-z_{2} \Vert _{E_{\epsilon}}. \end{aligned}$$

• 3. For any $z\in B_{\epsilon}$, there hold

  $$\begin{aligned} & \bigl\Vert S_{\epsilon}^{m} z- {\mathcal {L}} S_{0}^{m}\Pi_{\epsilon} z \bigr\Vert _{E_{1}} \leq\Upsilon_{2}^{m} \epsilon^{\beta}, \quad\forall m \in\mathbb{N}, \\ & \bigl\Vert S_{\epsilon}(t) z- {\mathcal {L}} S_{0}(t) \Pi_{\epsilon} z \bigr\Vert _{E_{1}} \leq\Upsilon_{3} \epsilon^{\gamma}, \quad\forall t\in\bigl[t^{\star}, 2t^{\star}\bigr]. \end{aligned}$$

  Here the “lifting” map ${\mathcal {L}}:B_{0}\rightarrow E_{\epsilon}$ is defined by

  $$ {\mathcal {L}} x= \textstyle\begin{cases} (x,{\mathscr {L}}x),&\textit{if }\epsilon>0,\\ x,&\textit{if }\epsilon=0, \end{cases} $$

  and $\Pi_{\epsilon}:B_{\epsilon}\rightarrow B_{0}$ is the projection onto the first component when $\epsilon>0$, and the identity map otherwise.

• 4. The map $z\mapsto S_{\epsilon}(t)z$ is Lipschitz continuous on $B_{\epsilon}$ endowed with the metric topology of $E_{\epsilon}$, with a Lipschitz constant independent of ϵ and $t\in[t^{\star}, 2t^{\star}]$.

• 5. The map

  $$ (t,z)\mapsto S_{\epsilon}(t)z:\bigl[t^{\star}, 2t^{\star}\bigr]\times B_{\epsilon}\rightarrow B_{\epsilon}$$

  is Hölder continuous of exponent ϑ, where $B_{\epsilon}$ is endowed with the metric topology of $E_{\epsilon}$.

Then there exists a family of exponential attractors ${\mathcal {E}}_{\epsilon}$ on ${\mathcal {B}}_{\epsilon}=\overline{B_{\epsilon}}^{E_{\epsilon}}$ with the following properties:

• (i) ${\mathcal {E}}_{\epsilon}$ attracts ${\mathcal {B}}_{\epsilon}$ with an exponential rate which is uniform with respect to ϵ, that is,

  $$ \operatorname{dist}_{E_{\epsilon}}\bigl(S_{\epsilon}(t) { \mathcal {B}}_{\epsilon}, {\mathcal {E}}_{\epsilon}\bigr)\leq M_{1} e^{-\omega t}, \quad\forall t\geq0, $$

  (3.1)

  for some $M_{1}>0$ and some $\omega>0$.

• (ii) The fractal dimension of ${\mathcal {E}}_{\epsilon}$ (denoted as $\dim_{F}({\mathcal {E}}_{\epsilon})$) is uniformly bounded with respect to ϵ, that is,

  $$ \dim_{F}({\mathcal {E}}_{\epsilon}) \leq M_{2}. $$

  (3.2)

• (iii) The family ${\mathcal {E}}_{\epsilon}$ is Hölder continuous with respect to ϵ, that is, there exist a positive constant $M_{3}$ and $\tau\in(0,\frac{1}{2}]$ such that

  $$ \operatorname{dist}_{E_{\epsilon}}^{\mathrm{sym}}( {\mathcal {E}}_{\epsilon}, {\mathcal {L}} {\mathcal {E}}_{0} ) \leq M_{3} \epsilon^{\tau}, $$

  (3.3)

  for all $0 <\epsilon\leq1$. In addition, there exist a positive constant $M_{4}$ and $\sigma\in(0,\frac{1}{2}]$ such that

  $$ \operatorname{dist}_{E_{1}}( {\mathcal {E}}_{\epsilon},{ \mathcal {L}} {\mathcal {E}}_{0} ) \leq M_{4} \epsilon^{\sigma}, $$

  (3.4)

  for all $0< \epsilon\leq1$, and

  $$ \lim_{\epsilon\to0}\operatorname{dist}_{E_{1}}( { \mathcal {L}} {\mathcal {E}}_{0}, {\mathcal {E}}_{\epsilon} )=0. $$

  (3.5)

Here ω, τ, σ and $M_{j}$ are independent of ϵ, and they can be computed explicitly.

We observe that the solution to the unperturbed problem (i.e., when $\epsilon=0$ in (1.1)) for the pair $(\phi, u)$ at any time t is given by $(\phi(t),u(t))=S(t)(\phi_{0},u_{0})$ and $\phi_{t}={\mathscr {L}}(\phi (t),u(t))$, where

$$\begin{aligned} {\mathscr {L}}(\varphi,\vartheta)=-\bigl(-\Delta\varphi+\varphi-g( \varphi )-\vartheta\bigr). \end{aligned}$$

(3.6)

Let $z_{1},z_{2}\in\mathcal{B}_{2}$, $z_{1}=(\phi^{1}_{0},\phi^{1}_{1},u^{1}_{0})$ and $z_{2}=(\phi^{2}_{0},\phi^{2}_{1},u^{2}_{0})$ be initial data for two solutions $(\phi^{1},u^{1})$ and $(\phi^{2},u^{2})$ of (1.1), respectively.

We set $(\phi(t),\phi_{t}(t),u(t))=S_{\epsilon}(t)z_{1}-S_{\epsilon}(t)z_{2}$, $\tilde{\phi}_{0}=\phi_{0}^{1}-\phi_{0}^{2}$, $\tilde{\phi}_{1}=\phi _{1}^{1}-\phi_{1}^{2}$ and $\tilde{u}_{0} =u_{0}^{1}-u_{0}^{2}$. Furthermore, we perform the splitting

$$\begin{aligned} &\bigl(\phi(t),\phi_{t}(t),u(t)\bigr)=\bigl(\chi(t), \chi_{t}(t),\vartheta(t)\bigr)+\bigl( \Psi (t),\Psi_{t}(t), \upsilon(t)\bigr), \end{aligned}$$

where $K_{\epsilon}(z_{1},z_{2})=(\chi(t),\chi_{t}(t),\vartheta(t))$ and $L_{\epsilon}(z_{1},z_{2})=(\Psi(t),\Psi_{t}(t),\upsilon(t))$ respectively solve the problems:

$$ \textstyle\begin{cases} \epsilon\chi_{tt}+\chi_{t}-\Delta\chi_{t}+\chi+ g(\phi_{1})-g(\phi _{2})- \vartheta=0,\\ \vartheta_{t}+\chi_{t}-\Delta\vartheta=0,\\ \chi|_{t=0}=0,\qquad \chi_{t}{|_{t=0}=0},\qquad \vartheta|_{t=0}=0 \end{cases} $$

(3.7)

and

$$ \textstyle\begin{cases} \epsilon\Psi_{tt}+\Psi_{t}-\Delta\Psi+ \Psi- \upsilon=0,\\ \upsilon_{t}+ \Psi_{t}-\Delta\upsilon=0,\\ \Psi|_{t=0}= \tilde{\phi}_{0},\qquad \Psi_{t}{|_{t=0}}= \tilde{\phi}_{1},\qquad \upsilon|_{t=0}=\tilde{u}_{0}. \end{cases} $$

(3.8)

Proposition 3.1

There exist $c,c',c_{1}>0$ independent of ϵ such that

$$\begin{aligned} & \bigl\Vert L_{\epsilon}(z_{1},z_{2}) \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}}\leq ce^{-c_{1}t} \Vert z_{1}-z_{2} \Vert _{{\mathcal {H}}_{1,\epsilon}},\quad\forall t\geq0,\quad \textit{and} \end{aligned}$$

(3.9)

$$\begin{aligned} & \bigl\Vert K_{\epsilon}(z_{1},z_{2}) \bigr\Vert _{{\mathcal {H}}_{2,\epsilon}}\leq ce^{c't} \Vert z_{1}-z_{2} \Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}},\quad\forall t\geq0. \end{aligned}$$

(3.10)

Proof

Firstly, we multiply (3.8)₁ by $\Psi_{t}$ and (3.8)₂ by υ, integrate over Ω, then add the resulting equations to get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla\Psi \Vert ^{2}+ \Vert \Psi \Vert ^{2}+\epsilon \Vert \Psi_{t} \Vert ^{2}+ \Vert \upsilon \Vert ^{2} \bigr)+ \Vert \Psi_{t} \Vert ^{2}+ \Vert \nabla\upsilon \Vert ^{2}=0. \end{aligned}$$

(3.11)

Next, we multiply (3.8)₁ by Ψ to obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl[ \Vert \Psi \Vert ^{2}+2\epsilon(\Psi,\Psi _{t}) \bigr]-\epsilon \Vert \Psi_{t} \Vert ^{2}+ \Vert \nabla\Psi \Vert ^{2}+ \Vert \Psi \Vert ^{2}-(\upsilon,\Psi)=0, \end{aligned}$$

and then deduce that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl[ \Vert \Psi \Vert ^{2}+2\epsilon(\Psi,\Psi _{t}) \bigr]+ \Vert \nabla\Psi \Vert ^{2}+\frac{1}{2} \Vert \Psi \Vert ^{2}+2 \epsilon(\Psi _{t},\Psi)\leq5\epsilon \Vert \Psi_{t} \Vert ^{2}+c \Vert \nabla\upsilon \Vert ^{2}. \end{aligned}$$

(3.12)

Summing (3.11) and κ (3.12), for some $\kappa\in(0,1)$ small enough, we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla\Psi \Vert ^{2}+(1+\kappa) \Vert \Psi \Vert ^{2}+\epsilon \Vert \Psi_{t} \Vert ^{2}+ \Vert \upsilon \Vert ^{2}+2\kappa\epsilon(\Psi,\Psi _{t}) \bigr)+\kappa \Vert \nabla\Psi \Vert ^{2}+\frac{\kappa}{2} \Vert \Psi \Vert ^{2} \\ &\quad {}+\epsilon(1-5\kappa) \Vert \Psi_{t} \Vert ^{2}+(1-c \kappa) \Vert \nabla\upsilon \Vert ^{2}+2\kappa\epsilon(\Psi, \Psi_{t})\leq0. \end{aligned}$$

Hence, there exists a $c_{1}>0$ (independent of ϵ) such that

$$\begin{aligned} \frac{d}{dt}E_{2}(t)+c_{1}E_{2}(t)\leq0, \end{aligned}$$

where $E_{2}(t)=\|\nabla\Psi\|^{2}+(1+\kappa)\|\Psi\|^{2}+\epsilon\|\Psi_{t}\| ^{2}+\|\upsilon\|^{2}+2\kappa\epsilon(\Psi,\Psi_{t})$. Simple integration over $(0,t)$ gives

$$\begin{aligned} E_{2}(t)\leq e^{-c_{1}t}E_{2}(0),\quad \forall t\geq0. \end{aligned}$$

(3.13)

Clearly, by Young’s inequality, there exist $b_{3},b_{4}>0$ (independent of ϵ) such that

$$\begin{aligned} b_{3} \bigl\Vert (\Psi,\Psi_{t},\upsilon) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}\leq E_{2}(t)\leq b_{4} \bigl\Vert (\Psi,\Psi_{t},\upsilon) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}. \end{aligned}$$

(3.14)

It follows from (3.13) and (3.14) that

$$\begin{aligned} \bigl\Vert (\Psi,\Psi_{t},\upsilon) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}\leq e^{-c_{1}t} \bigl\Vert (\tilde{\phi}_{0},\tilde{\phi}_{1},\tilde{u}_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}},\quad\forall t\geq0. \end{aligned}$$

Hence (3.9) follows.

Secondly, we multiply (3.7)₁ by $\chi_{t}$ and (3.7)₂ by ϑ, integrate over Ω, then add the resulting equations to get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla\chi \Vert ^{2}+ \Vert \chi \Vert ^{2}+\epsilon \Vert \chi_{t} \Vert ^{2}+ \Vert \vartheta \Vert ^{2} \bigr)+ \Vert \chi_{t} \Vert ^{2}+ \Vert \nabla\vartheta \Vert ^{2}=-\bigl(g\bigl(\phi^{1}\bigr)-g \bigl(\phi^{2}\bigr),\chi_{t}\bigr). \end{aligned}$$

We have that $\|g(\phi^{1})-g(\phi^{2})\|\leq\|g'(\theta\phi^{1}+(1-\theta)\phi^{2})\| _{L^{\infty}(\Omega)}\|\phi\|$, where $\theta\in[0,1]$. It follows that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl( \Vert \nabla\chi \Vert ^{2}+ \Vert \chi \Vert ^{2}+\epsilon \Vert \chi_{t} \Vert ^{2}+ \Vert \vartheta \Vert ^{2} \bigr)+\frac{1}{2} \Vert \chi_{t} \Vert ^{2}+ \Vert \nabla\vartheta \Vert ^{2}\leq \Vert \phi \Vert ^{2}. \end{aligned}$$

(3.15)

Integrating (3.15) over $(0,t)$ and then accounting for (2.6), we deduce that

$$\begin{aligned} \Vert \chi \Vert _{1}^{2}+\epsilon \Vert \chi_{t} \Vert ^{2}+ \Vert \vartheta \Vert ^{2}\leq ce^{c't} \bigl\Vert (\tilde{\phi}_{0}, \tilde{\phi}_{1},\tilde{u}_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}, \quad\forall t\geq0. \end{aligned}$$

(3.16)

Next, we multiply (3.7)₁ by $-\Delta\chi_{t}$ and (3.7)₂ by Nϑ, integrate over Ω, then add the resulting equations to get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \Vert \Delta\chi \Vert ^{2}+ \Vert \nabla\chi \Vert ^{2}+\epsilon \Vert \nabla \chi_{t} \Vert ^{2}+ \Vert \nabla\vartheta \Vert ^{2} \bigr)+ \Vert \nabla\chi_{t} \Vert ^{2}+ \Vert \Delta\vartheta \Vert ^{2} \\ &\quad =-\bigl(\nabla\bigl(g\bigl(\phi^{1}\bigr)-g\bigl(\phi^{2} \bigr)\bigr),\nabla\chi_{t}\bigr). \end{aligned}$$

We have that $\|\nabla(g(\phi^{1})-g(\phi^{2}))\|\leq c\|\phi\|_{1}$. It follows that

$$\begin{aligned} &\frac{1}{2} \frac{d}{dt} \bigl( \Vert \Delta\chi \Vert ^{2}+ \Vert \nabla\chi \Vert ^{2}+\epsilon \Vert \nabla\chi_{t} \Vert ^{2}+ \Vert \nabla\vartheta \Vert ^{2} \bigr)+\frac {1}{2} \Vert \nabla\chi_{t} \Vert ^{2}+ \Vert \Delta\vartheta \Vert ^{2}\leq c \Vert \phi \Vert _{1}^{2}. \end{aligned}$$

(3.17)

Integrating (3.17) over $(0,t)$ and taking into account (2.6), we deduce that

$$\begin{aligned} \Vert \Delta\chi \Vert ^{2}+ \Vert \nabla\chi \Vert ^{2}+\epsilon \Vert \nabla\chi_{t} \Vert ^{2}+ \Vert \nabla\vartheta \Vert ^{2}\leq ce^{c't} \bigl\Vert (\tilde{\phi}_{0},\tilde{\phi}_{1},\tilde{u}_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}, \quad\forall t\geq0. \end{aligned}$$

(3.18)

On account of (3.16) and (3.18), we obtain that

$$\begin{aligned} \bigl\Vert (\chi,\chi_{t},\vartheta) \bigr\Vert ^{2}_{{\mathcal {H}}_{2,\epsilon}}\leq ce^{c't} \bigl\Vert (\tilde{\phi}_{0},\tilde{\phi}_{1},\tilde{u}_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}},\quad\forall t\geq0. \end{aligned}$$

Hence (3.10) follows. □

We prove the following result.

Theorem 3.2

For every $\epsilon\in(0,1]$, the semigroup ${S}_{\epsilon}(t)$ possesses an exponential attractor ${\mathcal {E}}_{\epsilon}$ (with dimension independent of ϵ) in ${\mathcal {H}}_{1,\epsilon}$.

Proof

Let $t\in[t^{*},2t^{*}]$ and set $(\phi(t),\phi_{t}(t),u(u))=S_{\epsilon}(t)z_{01}-S_{\epsilon}(t)z_{02}=(\phi^{1}(t),\phi^{1}_{t}(t),u^{1}(t))-(\phi ^{2}(t),\phi_{t}^{2}(t),u^{2}(t))$. Therefore, the triplet $(\phi(t),\phi _{t}(t),u(u))$ is a solution to the problem

$$ \textstyle\begin{cases} \epsilon\phi_{tt}+\phi_{t}-\Delta\phi+ \phi+g(\phi_{1})-g(\phi_{2}) - u=0,\\ u_{t}+\phi_{t}-\Delta u =0,\\ \phi|_{t=0}= \phi^{01}-\phi^{02}, \qquad\phi_{t}{|_{t=0}}= \phi ^{01}_{1}-\phi_{1}^{02}, \qquad u|_{t=0}=u^{01}-u^{02}. \end{cases} $$

(3.19)

On account of (2.6) we obtain

$$ \bigl\Vert S_{\epsilon}(t)z_{01}-S_{\epsilon}(t)z_{02} \bigr\Vert _{\mathcal{H}_{1,\epsilon }}\leq c\bigl(t^{*}\bigr) \Vert z_{01}-z_{02} \Vert _{\mathcal{H}_{1,\epsilon}}, \quad t\leq2t^{*}, $$

(3.20)

where $c(t^{*})>0$ is independent of ϵ. Now, we multiply (1.1)₁ and (1.1)₂ by $-\Delta\phi_{t}$ and $-\Delta u$, respectively, integrate over Ω then add the resulting equations, and deduce

$$\begin{aligned} &\frac{d}{dt} \bigl( \Vert \Delta\phi \Vert ^{2}+ \Vert \nabla\phi \Vert ^{2}+\epsilon \Vert \nabla\phi_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2} \bigr)+ \Vert \nabla\phi_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \\ &\quad \leq\frac{1}{2} \bigl\Vert g'(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \nabla\phi \Vert ^{2} \\ &\quad \leq c \Vert \nabla\phi \Vert ^{2}. \end{aligned}$$

Integrating over $(0,t)$ and recalling (2.2), we get

$$\begin{aligned} & \Vert \Delta\phi \Vert ^{2}+ \Vert \nabla\phi \Vert ^{2}+\epsilon \Vert \nabla\phi_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \int_{0}^{t}\bigl( \bigl\Vert \nabla \phi_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \Delta u(s) \bigr\Vert ^{2}\bigr)\,ds \\ &\quad \leq c(t+1),\quad\forall t\geq0. \end{aligned}$$

(3.21)

It then follows from (2.3) and (3.21) that

$$\begin{aligned} \int_{0}^{t}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert _{1}^{2}+ \bigl\Vert \Delta u(s) \bigr\Vert ^{2}\bigr)\,ds\leq c(t+1),\quad \forall t\geq0. \end{aligned}$$

(3.22)

Next, from (1.1)₁, we deduce that

$$\begin{aligned} \epsilon^{2} \int_{0}^{t} \bigl\Vert \phi_{tt}(s) \bigr\Vert ^{2}\,ds\leq \int_{0}^{t}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \Delta\phi(s) \bigr\Vert ^{2}+ \bigl\Vert \phi(s) \bigr\Vert ^{2}+ \bigl\Vert g \bigl(\phi(s)\bigr) \bigr\Vert ^{2}+ \bigl\Vert u(s) \bigr\Vert ^{2}\bigr)\,ds, \end{aligned}$$

then from (2.2), (3.21) and (3.22) it follows that

$$\begin{aligned} \int_{0}^{t}\epsilon \bigl\Vert \phi_{tt}(s) \bigr\Vert ^{2}\leq\frac{c}{\epsilon }(t+1), \quad\forall t\geq0. \end{aligned}$$

(3.23)

Also, from (1.1)₂ and (3.22), we deduce that

$$\begin{aligned} \int_{0}^{t} \bigl\Vert u_{t}(s) \bigr\Vert ^{2}\,ds&\leq c \int_{0}^{t}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \Delta u(s) \bigr\Vert ^{2} \bigr)\,ds \\ &\leq c(t+1),\quad\forall t\geq0. \end{aligned}$$

(3.24)

Finally, we have that

$$\begin{aligned} & \bigl\Vert {S}_{\epsilon}(t)z_{01}-{S}_{\epsilon}\bigl(t'\bigr)z_{02} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}} \\ &\quad \leq \bigl\Vert {S}_{\epsilon}(t)z_{01}-{S}_{\epsilon}\bigl(t'\bigr)z_{01} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}}+ \bigl\Vert {S}_{\epsilon}\bigl(t'\bigr)z_{01}-{S}_{\epsilon}\bigl(t'\bigr)z_{02} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}}, \quad \forall t, t' \in\bigl[t^{*},2t^{*}\bigr]. \end{aligned}$$

Indeed, on the one hand, from (3.23) and (3.24), we have

$$\begin{aligned} & \bigl\Vert {S}_{\epsilon}(t)z_{01}-{S}_{\epsilon}\bigl(t'\bigr)z_{01} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}} \\ &\quad \leq c \bigl( \bigl\Vert \phi(t)-\phi\bigl(t'\bigr) \bigr\Vert _{1}+\sqrt{\epsilon}\bigl\Vert \phi_{t}(t)-\phi _{t}\bigl(t'\bigr) \bigr\Vert + \bigl\Vert u(t)-u \bigl(t'\bigr) \bigr\Vert \bigr) \\ &\quad \leq c \int_{t}^{t'}\bigl( \bigl\Vert \phi_{t}(s) \bigr\Vert _{1}+\sqrt{\epsilon}\bigl\Vert \phi_{tt}(s) \bigr\Vert + \bigl\Vert u_{t}(s) \bigr\Vert \bigr)\,ds \\ &\quad \leq c\bigl( \epsilon,t^{*}\bigr) \bigl\vert t'-t \bigr\vert ^{1/2}. \end{aligned}$$

On the other hand, it follows from (3.20) that

$$\begin{aligned} \bigl\Vert {S}_{\epsilon}\bigl(t' \bigr)z_{01}-{S}_{\epsilon}\bigl(t' \bigr)z_{02} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}} \leq c\bigl(t^{*}\bigr) \Vert z_{01}-z_{02} \Vert _{{\mathcal {H}}_{1,\epsilon}}, \quad\forall t'\geq0. \end{aligned}$$

(3.25)

Hence, we conclude with

$$\begin{aligned} \bigl\Vert {S}_{\epsilon}(t)z_{01}-{S}_{\epsilon}\bigl(t'\bigr)z_{02} \bigr\Vert _{{\mathcal {H}}_{1,\epsilon}} \leq c \bigl(\epsilon,t^{*}\bigr) \bigl( \bigl\vert t'-t \bigr\vert ^{1/2} + \Vert z_{01}-z_{02} \Vert _{{\mathcal {H}}_{1,\epsilon}}\bigr). \end{aligned}$$

(3.26)

We now apply Theorem 3.1. We will only need to check Assumptions 2, 4 and 5, for the existence of a family of exponential attractors ${\mathcal {E}}_{\epsilon}$ that satisfy (3.1) and (3.2). Assumption 2 follows from estimates (3.9) and (3.10) of Proposition 3.1. Assumptions 4 and 5 follow from (2.6) and (3.26), respectively. This shows the existence of a family of exponential attractors ${\mathcal {E}}_{\epsilon}$ in ${\mathcal {H}}_{1,\epsilon}$ with dimension independent of ϵ. □

4 Robust family of exponential attractors

We start by showing the existence of an absorbing set in ${\mathcal {H}}_{3,\epsilon}$.

Proposition 4.1

The semigroup $S_{\epsilon}(t)$ possesses an exponentially attracting bounded absorbing set ${\mathcal {B}}_{3}$ in ${\mathcal {H}}_{3,\epsilon}$.

Proof

Let $B\subset{\mathcal {H}}_{3,\epsilon}$ be a bounded set, and let $(\phi_{0},\phi_{1},u_{0})\in B$. Hence, since ${\mathcal {H}}_{3,\epsilon }\subset{\mathcal {H}}_{2,\epsilon}$, there exists a $t(B)>0$ such that $(\phi(t),\phi_{t}(t),u(t))\in{\mathcal {B}}_{2}$, $\forall t\geq t(B)$. That is,

$$ \bigl\Vert \phi(t) \bigr\Vert _{2}^{2}+ \epsilon \bigl\Vert \phi_{t}(t) \bigr\Vert _{1}^{2}+ \bigl\Vert u(t) \bigr\Vert _{1}^{2}\leq r_{2}, \quad \forall t\geq t(B). $$

(4.1)

The following estimates hold true:

$$\begin{aligned}& \begin{aligned}[b]\bigl(\Delta g(\phi), \Delta\phi_{t}\bigr)&\leq \bigl\Vert g'(\phi) \bigr\Vert _{L^{\infty}(\Omega )} \Vert \Delta\phi \Vert \Vert \Delta\phi_{t} \Vert + \bigl\Vert g''(\phi) \bigr\Vert _{L^{\infty}(\Omega )} \Vert \nabla \phi \Vert ^{2}_{L^{4}{\Omega}} \Vert \Delta\phi_{t} \Vert \\ &\leq c\bigl( \bigl\Vert g'(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \Delta\phi \Vert ^{2}+ \bigl\Vert g''(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \nabla\phi \Vert _{1}^{4}\bigr)+\frac{1}{2} \Vert \Delta\phi_{t} \Vert ^{2}, \end{aligned} \end{aligned}$$

(4.2)

$$\begin{aligned}& \begin{aligned}[b] \bigl(g(\phi),\Delta^{2}\phi\bigr)&\leq \bigl\Vert g'(\phi) \bigr\Vert _{L^{\infty}(\Omega)} \Vert \nabla\phi \Vert \Vert \nabla\Delta\phi \Vert \\ &\leq \bigl\Vert g'(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \nabla\phi \Vert ^{2}+\frac {1}{4} \Vert \nabla\Delta \phi \Vert ^{2}, \end{aligned} \end{aligned}$$

(4.3)

$$\begin{aligned}& \bigl(u,\Delta^{2}\phi\bigr) \leq \Vert \nabla u \Vert ^{2}+\frac{1}{4} \Vert \nabla\Delta\phi \Vert ^{2}, \end{aligned}$$

(4.4)

$$\begin{aligned}& \epsilon(\Delta\phi,\Delta\phi_{t}) \leq \frac{1}{2} \Vert \Delta\phi \Vert ^{2}+\epsilon \Vert \Delta\phi_{t} \Vert ^{2}. \end{aligned}$$

(4.5)

Multiply (1.1)₁ by $\Delta^{2}\phi_{t}$ and $\kappa\Delta ^{2}\phi$ with $0<\kappa\leq\frac{1}{8}$, then multiply (1.1)₂ by $\Delta^{2}u$, and integrate over Ω. Adding the resulting equations gives, on account of (4.2)–(4.5),

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \Vert \nabla\Delta\phi \Vert ^{2}+(1+\kappa) \Vert \Delta\phi \Vert ^{2}+\epsilon \Vert \Delta\phi_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\kappa \epsilon(\Delta\phi,\Delta\phi_{t}) \bigr] \\ &\qquad {}+\frac{\kappa}{2}\|\nabla\Delta\phi|^{2}+\frac{\kappa}{2} \Vert \Delta \phi \Vert ^{2}+\epsilon \biggl(\frac{1}{2}-2\kappa \biggr) \Vert \Delta\phi _{t} \Vert ^{2}+\epsilon\kappa( \Delta\phi,\Delta\phi_{t}) \\ &\quad \leq c\bigl( \bigl\Vert g'(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \Delta\phi \Vert ^{2}+ \bigl\Vert g''(\phi) \bigr\Vert ^{2}_{L^{\infty}(\Omega)} \Vert \nabla\phi \Vert ^{4}_{1}+ \Vert \nabla u \Vert ^{2}\bigr). \end{aligned}$$

Hence from (4.1), there exists a constant $\varpi_{1}>0$ independent of ϵ such that

$$\begin{aligned} \frac{d}{dt}E_{3}(t)+\varpi_{1}E_{3}(t) \leq c(r_{2}), \end{aligned}$$

(4.6)

where

$$E_{3}(t)= \Vert \nabla\Delta\phi \Vert ^{2}+(1+\varpi) \Vert \Delta\phi \Vert ^{2}+\epsilon \Vert \Delta \phi_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\varpi\epsilon(\Delta\phi ,\Delta\phi_{t}). $$

Clearly, by Hölder’s and Young’s inequalities, there exist constants $\varpi_{2},\varpi_{3}>0$, independent of ϵ such that

$$\begin{aligned} &\varpi_{2}\bigl( \Vert \nabla\Delta\phi \Vert ^{2}+ \Vert \Delta\phi \Vert ^{2}+\epsilon \Vert \Delta \phi_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \bigr) \\ &\quad \leq E_{3}(t) \\ &\quad \leq\varpi_{3}\bigl( \Vert \nabla\Delta\phi \Vert ^{2}+ \Vert \Delta\phi \Vert ^{2}+\epsilon \Vert \Delta \phi_{t} \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \bigr). \end{aligned}$$

(4.7)

Applying the generalized Gronwall’s lemma to (4.6) and using (4.7), we obtain

$$\begin{aligned} \bigl\Vert \bigl(\phi(t),\phi_{t}(t),u(t)\bigr) \bigr\Vert ^{2}_{{\mathcal {H}}_{3,\epsilon}}\leq c(B)e^{-\varpi_{1}t}+c(r_{2}),\quad \forall t\geq0. \end{aligned}$$

(4.8)

Hence, we have that

$${\mathcal {B}}_{3}=\bigl\{ (\varphi,\psi,v)\in{\mathcal {H}}_{3,\epsilon}, \bigl\Vert (\varphi,\psi,v) \bigr\Vert _{{\mathcal {H}}_{3,\epsilon}} \leq\sqrt {{2c(r_{2})}/{\varpi_{1}}}=r_{3}\bigr\} $$

is an exponentially attracting absorbing set for $S_{\epsilon}(t)$ on ${\mathcal {H}}_{3.,\epsilon}$. □

We prove the following result.

Proposition 4.2

For every $\epsilon\in(0,1]$, there exists a $c>0$, independent of ϵ, such that for any ${\mathrm{z}}\in{\mathcal {B}}_{3}$,

$$\begin{aligned} \bigl\Vert S_{\epsilon}(t){\mathrm{z}} \bigr\Vert _{{\mathcal {H}}_{2,0}}\le c,\quad \forall t\ge1. \end{aligned}$$

(4.9)

Proof

Let ${\mathrm{z}}_{0}=(\phi_{0},\phi_{1},u_{0})\in{\mathcal {B}}_{3}$. We set $(\phi(t),\phi_{t}(t),u(t))=S_{\epsilon}(t)(\phi_{0},\phi_{1},u_{0})$, $\forall t\ge0$. There exists a $c>0$, independent of ϵ, such that

$$\begin{aligned} & \bigl\Vert \phi(t) \bigr\Vert _{3}^{2}+ \epsilon \bigl\Vert \phi_{t}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert u(t) \bigr\Vert ^{2}_{2}\le c,\quad \forall t \ge0. \end{aligned}$$

(4.10)

Multiplying the first equation of (1.1) by $\varGamma\phi_{t}$, where $\varGamma=I-\Delta$, then integrating over Ω, we obtain

$$\begin{aligned} \frac{\epsilon}{2}\frac{d}{dt} \Vert \phi_{t} \Vert _{1}^{2}+ \Vert \phi_{t} \Vert _{1}^{2}+(-\Delta\phi,\varGamma\phi_{t})+(\phi, \varGamma\phi _{t})+\bigl(g(\phi),\varGamma\phi_{t}\bigr)-(u, \varGamma\phi_{t})=0. \end{aligned}$$

Hence, we deduce due to (4.10), that

$$ \epsilon\frac{d}{dt} \Vert \phi_{t} \Vert _{1}^{2}+ \Vert \phi_{t} \Vert _{1}^{2}\leq c. $$

(4.11)

First, we multiply (4.11) by $e^{ct/\epsilon}$ and integrate between τ and $t+1$, for any $\tau\le t+1$. This yields

$$\begin{aligned} \epsilon \bigl\Vert \phi_{t}(t+1) \bigr\Vert _{1}^{2}e^{c(t+1)/\epsilon}\le c\epsilon \bigl\Vert \phi _{t}(\tau) \bigr\Vert _{1}^{2}e^{cs/\epsilon}+c \epsilon \bigl( e^{c(t+1)/\epsilon }-e^{c\tau/\epsilon} \bigr). \end{aligned}$$

(4.12)

Now, integrating (4.12) between t and $t+1$ with respect to τ, we deduce

$$\begin{aligned} \bigl\Vert \phi_{t}(t) \bigr\Vert _{1}^{2} \le c,\quad\forall t\ge1, \end{aligned}$$

(4.13)

hence the estimate (4.9) holds. □

The following estimate holds for difference of two solutions.

Proposition 4.3

There exist $t_{\star}>0$, c and $c'>0$ all independent of ϵ such that

$$\begin{aligned} \bigl\Vert S_{\epsilon}(t) (\phi_{0}, \phi_{1},u_{0})-{\mathcal {L}}S(t) (\phi_{0},u_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,\epsilon}}\leq c\sqrt[4]{\epsilon}e^{c't}, \quad \forall t\geq t_{\star}, \end{aligned}$$

(4.14)

for any $(\phi_{0},\phi_{1},u_{0})\in{\mathcal {B}}_{3}$, and

$$\begin{aligned} \bigl\Vert S_{\epsilon}(t) (\phi_{0}, \phi_{1},u_{0})-{\mathcal {L}}S(t) (\phi_{0},u_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,0}}\leq c\sqrt[4]{\epsilon}e^{c't}, \quad\forall t\geq t_{\star}, \end{aligned}$$

(4.15)

for any $(\phi_{0},\phi_{1},u_{0})\in S_{\epsilon}(1){\mathcal {B}}_{3}$, and any $\epsilon\in(0,1]$, where ${\mathcal {L}}(\psi(t),\upsilon(t))=(\psi(t), {\mathscr {L}}(\psi (t), \upsilon(t)), \upsilon(t))$.

Proof

Let $(\phi_{0},\phi_{1},u_{0})\in{\mathcal {B}}_{3}$. We set $(\phi^{\epsilon}(t),\phi^{\epsilon}_{t}(t),u^{\epsilon}(t))=S_{\epsilon}(t)(\phi_{0},\phi_{1},u_{0})$, and $(\phi(t),\phi_{t}(t), u(t))={\mathcal {L}}S(t)(\phi_{0},u_{0})$.

We have that

$$\begin{aligned} & \bigl\Vert \phi^{\epsilon}(t) \bigr\Vert _{3}^{2}+ \epsilon \bigl\Vert \phi_{t}^{\epsilon}(t) \bigr\Vert _{2}^{2}+ \bigl\Vert u^{\epsilon}(t) \bigr\Vert _{2}^{2}\leq c,\quad\forall t\geq0, \end{aligned}$$

(4.16)

$$\begin{aligned} & \bigl\Vert \phi(t) \bigr\Vert _{3}^{2}+ \bigl\Vert u(t) \bigr\Vert _{2}^{2}\leq c, \quad\forall t \geq0. \end{aligned}$$

(4.17)

We set $P=\phi^{\epsilon}-\phi$ and $R=u^{\epsilon}-u$, then the pair $(P,R)$ solves the problem:

$$ \textstyle\begin{cases} \epsilon P_{tt}+P_{t}-\Delta P+ P+g(\phi^{\epsilon})-g(\phi) - R=-\epsilon\phi_{tt},\\ R_{t}+P_{t}-\Delta R =0,\\ P|_{t=0}= 0, \qquad P_{t}{|_{t=0}}= \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}),\qquad R|_{t=0}=0. \end{cases} $$

(4.18)

We multiply (4.18)₁ and (4.18)₁ by $P_{t}$ and R, respectively, then integrate over Ω. Adding the resulting equations, we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \bigl( \Vert P \Vert _{1}^{2}+\epsilon \Vert P_{t} \Vert ^{2}+ \Vert R \Vert ^{2} \bigr)+ \Vert P_{t} \Vert + \Vert \nabla R \Vert ^{2} =-\bigl(g\bigl( \phi^{\epsilon}\bigr)-g(\phi), P_{t}\bigr)-\epsilon( \phi_{tt},P_{t}). \end{aligned}$$

We deduce that

$$\begin{aligned} \frac{d}{dt} \bigl( \Vert P \Vert _{1}^{2}+ \epsilon \Vert P_{t} \Vert ^{2}+ \Vert R \Vert ^{2} \bigr)+ \Vert P_{t} \Vert ^{2}+ \Vert \nabla R \Vert ^{2} \le c' \Vert P \Vert ^{2}+2\epsilon^{2} \Vert \phi_{tt} \Vert ^{2}. \end{aligned}$$

(4.19)

The following holds true:

$$ \int_{0}^{t} \bigl\Vert \phi_{tt}(s) \bigr\Vert ^{2}\,ds\leq c e^{\nu t}, \quad\forall t\geq0. $$

(4.20)

We integrate (4.19) over $(0,t)$, and on account of (4.20) we obtain

$$\begin{aligned} & \bigl\Vert P(t) \bigr\Vert _{1}^{2}+ \epsilon \bigl\Vert P_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert R(t) \bigr\Vert ^{2}\leq c \bigl(\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert ^{2}+\epsilon^{2} \bigr) e^{c't}, \quad\forall t \geq0. \end{aligned}$$

(4.21)

Similarly, we multiply (4.18)₁ and (4.18)₁ by $-\Delta P_{t}$ and $-\Delta R$, respectively, then integrate over Ω. Adding the resulting equations and proceeding like in the proof of estimate (4.21) above, we obtain

$$\begin{aligned} & \bigl\Vert P(t) \bigr\Vert _{2}^{2}+ \epsilon \bigl\Vert \nabla P_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert R(t) \bigr\Vert _{1}^{2}\leq c \bigl(\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}+\epsilon^{2} \bigr) e^{c't}, \quad\forall t\geq0. \end{aligned}$$

(4.22)

Now, integrating (4.19) between 0 and t, we obtain

$$\begin{aligned} & \int_{0}^{t} \bigl( \bigl\Vert P_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert R(s) \bigr\Vert _{1}^{2} \bigr) \,ds\leq c \bigl(\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}( \phi_{0},u_{0}) \bigr\Vert ^{2}+ \epsilon^{2} \bigr) e^{c't}, \quad\forall t\geq0, \end{aligned}$$

(4.23)

due to (4.20) and (4.21). Next, we multiply (4.18)₁ by $P_{t}$ and integrate over Ω to deduce

$$ \frac{d}{dt}\epsilon \Vert P_{t} \Vert ^{2}+ \Vert P_{t} \Vert ^{2}\leq c\bigl( \Vert P \Vert _{2}^{2}+ \Vert R \Vert ^{2}+ \epsilon^{2} \Vert \phi_{tt} \Vert ^{2}\bigr). $$

(4.24)

We multiply (4.24) by t to get

$$\begin{aligned} \frac{d}{dt} \bigl(\epsilon t \Vert P_{t} \Vert ^{2}e^{t/\epsilon} \bigr)\leq \epsilon \Vert P_{t} \Vert ^{2}e^{t/\epsilon}+ \bigl[ct\bigl( \Vert P \Vert ^{2}+ \Vert R \Vert ^{2}+\epsilon^{2} \Vert \phi_{tt} \Vert ^{2}\bigr) \bigr]e^{t/\epsilon}. \end{aligned}$$

(4.25)

Integrating (4.25) between 0 and t, due to (4.20), (4.21), (4.22) and (4.23), we obtain

$$\begin{aligned} \epsilon t \bigl\Vert P_{t}(t) \bigr\Vert ^{2} &\leq \epsilon \int_{0}^{t} \bigl\Vert P_{t}(s) \bigr\Vert ^{2}\,ds+ c \epsilon t \bigl(\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}+\epsilon^{2} \bigr)e^{c't} \\ &\quad{} +c\epsilon^{2}t \int_{0}^{t} \bigl\Vert \phi_{tt}(s) \bigr\Vert ^{2}\,ds \\ &\leq c\epsilon\bigl(\epsilon^{2}+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}) \bigr\Vert ^{2}\bigr)e^{c't}+ct\epsilon\bigl(\epsilon^{2}+ \epsilon \bigl\Vert \phi _{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}\bigr)e^{c't} \\ &\quad{} +ct\epsilon^{2}e^{c't}, \quad\forall t\geq0. \end{aligned}$$

Hence

$$\begin{aligned} \epsilon \bigl\Vert P_{t}(t) \bigr\Vert ^{2}&\leq c \epsilon t^{-1}\bigl(\epsilon^{2}+\epsilon \bigl\Vert \phi _{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert ^{2}\bigr)e^{c't} \\ &\quad{} +c\epsilon\bigl(\epsilon+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}) \bigr\Vert _{1}^{2} \bigr)e^{c't},\quad\forall t\geq0. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \epsilon \bigl\Vert P_{t}(\sqrt{\epsilon}) \bigr\Vert ^{2} &\leq c\sqrt{\epsilon }\bigl(\epsilon^{2}+ \epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert ^{2}\bigr) +c\epsilon\bigl(\epsilon+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}\bigr). \end{aligned}$$

(4.26)

Using the interpolation inequality, (4.22) and (4.23), we deduce

$$\begin{aligned} \bigl\Vert P(t) \bigr\Vert _{1}^{2}&\leq c \bigl\Vert P(t) \bigr\Vert \bigl\Vert P(t) \bigr\Vert _{2} \\ &\leq c\sqrt{t}\bigl(\epsilon^{2}+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}) \bigr\Vert _{1}^{2}\bigr)e^{c't}, \quad\forall t\geq0. \end{aligned}$$

Therefore,

$$\begin{aligned} \bigl\Vert P(\sqrt{\epsilon}) \bigr\Vert _{1}^{2} \leq c\sqrt[4]{\epsilon}\bigl(\epsilon ^{2}+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}\bigr). \end{aligned}$$

(4.27)

From (4.18)₂ and (4.23), we deduce

$$\begin{aligned} \int_{0}^{t} \bigl\Vert R_{t}(s) \bigr\Vert _{-1}^{2}\,ds&\leq c \int_{0}^{t}\bigl( \bigl\Vert P_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla R(s) \bigr\Vert ^{2} \bigr)\,ds \\ &\leq c\bigl(\epsilon^{2}+\epsilon \bigl\Vert \phi_{1}-{ \mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert ^{2} \bigr)e^{c't},\quad\forall t\geq0. \end{aligned}$$

(4.28)

Again, by interpolation inequality, (4.22) and (4.28), we have

$$\begin{aligned} \bigl\Vert R(t) \bigr\Vert ^{2}&\le c \bigl\Vert R(t) \bigr\Vert _{-1} \bigl\Vert R(t) \bigr\Vert _{1} \\ &\le c\sqrt {t} \bigl(\epsilon^{2}+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}) \bigr\Vert ^{2}_{1} \bigr) e^{c't}, \quad\forall t\ge 0, \end{aligned}$$

so that

$$\begin{aligned} \bigl\Vert R(\sqrt{\epsilon}) \bigr\Vert ^{2}\le c \sqrt[4]{\epsilon} \bigl(\epsilon ^{2}+ \epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2} \bigr). \end{aligned}$$

(4.29)

We now apply Gronwall’s lemma to (4.19) between $\sqrt{\epsilon}$ and $t+\sqrt{\epsilon}$. We find

$$\begin{aligned} &\bigl( \Vert P \Vert _{1}^{2}+\epsilon \Vert P_{t} \Vert ^{2}+ \Vert R \Vert ^{2} \bigr) (t+\sqrt{\epsilon})\leq c \bigl[\bigl( \Vert P \Vert _{1}^{2}+ \epsilon \Vert P_{t} \Vert ^{2}+ \Vert R \Vert ^{2} \bigr) (\sqrt{\epsilon})+ \epsilon^{2} \bigr]e^{c't}, \end{aligned}$$

(4.30)

for every $t\ge0$.

Due to (4.26), (4.27) and (4.29), from (4.30) it follows that

$$\begin{aligned} &\bigl( \Vert P \Vert _{1}^{2}+\epsilon \Vert P_{t} \Vert ^{2}+ \Vert R \Vert ^{2} \bigr) (t+\sqrt{\epsilon})\leq c\sqrt[4]{\epsilon}\bigl(\epsilon+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi _{0},u_{0}) \bigr\Vert _{1}^{2}\bigr)e^{c't},\quad\forall t\ge0. \end{aligned}$$

(4.31)

Again, integrating (4.19) between s and t, we arrive at the following estimate:

$$ \bigl\Vert P(t) \bigr\Vert _{1}^{2}+\epsilon \bigl\Vert P_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert R(t) \bigr\Vert ^{2}\leq c\bigl( \bigl\Vert P(s) \bigr\Vert _{1}^{2}+\epsilon \bigl\Vert P_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert R(s) \bigr\Vert ^{2}+ \epsilon^{2}\bigr)e^{c't}, $$

for any given $s\geq0$ and any $t>s$. Let $t_{\star}>0$, independent of ϵ, be such that $t_{\star}>\sqrt{\epsilon}$. This latter estimate, with $s= \sqrt{\epsilon}$, in combination with (4.31) gives

$$\begin{aligned} & \bigl\Vert P(t) \bigr\Vert _{1}^{2}+ \epsilon \bigl\Vert P_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert R(t) \bigr\Vert ^{2} \leq c\sqrt [4]{\epsilon}\bigl(\epsilon+\epsilon \bigl\Vert \phi_{1}-{\mathscr {L}}(\phi_{0},u_{0}) \bigr\Vert _{1}^{2}\bigr)e^{c't}, \quad\forall t> \sqrt{\epsilon}. \end{aligned}$$

(4.32)

Finally, estimate (4.14) follows from (4.32) while estimate (4.15) follows from (4.9) and (4.32). □

We have the following corollary of Proposition 4.3.

Corollary 4.1

$$\begin{aligned} \bigl\Vert \Pi_{\epsilon}S_{\epsilon}(t) ( \phi_{0},\phi_{1},u_{0})-S(t) (\phi_{0},u_{0}) \bigr\Vert ^{2}_{{\mathcal {H}}_{1,0}}\leq c\sqrt[4]{\epsilon} e^{c't},\quad\forall t\geq t_{\star}, \end{aligned}$$

(4.33)

where $\Pi_{\epsilon}(X\times Y\times Z)=X\times Z$, i.e., $\|\phi^{\epsilon}(t)-\phi(t)\|_{1}^{2}+\|u^{\epsilon}(t)-u(t)\|^{2}\leq c\sqrt[4]{\epsilon} e^{c't}$, $\forall t\geq t_{\star}$.

The semigroup $S(t)$ for the variable $(\phi,u)$ alone possesses an exponential attractor ${\mathcal {E}}_{0}$ on ${\mathcal {H}}_{1,0}$, see Theorem 9.14 in [11]. We set $\widetilde {\mathcal {B}}_{3}=S_{\epsilon}(t^{*}){\mathcal {B}}_{3}$, where $t^{*}>0$ is independent of ϵ.

Theorem 4.1

There exist $\varpi_{1},\varpi_{2}\in(0, \frac{1}{2}]$ and $M_{1},M_{2}>0$, all independent of ϵ, and a family of exponential attractors ${\mathcal {E}}_{\epsilon}$ enjoying all the properties of Theorem 3.2 and such that

$$\begin{aligned} &\operatorname{dist}_{{\mathcal {H}}_{1,\epsilon }}^{\mathrm{sym}}({\mathcal {E}}_{\epsilon},{\mathcal {E}})\le M_{1} \epsilon ^{\varpi_{1}}, \end{aligned}$$

(4.34)

$$\begin{aligned} &\operatorname{dist}_{{\mathcal {H}}_{1,0}}({\mathcal {E}}_{\epsilon },{\mathcal {E}})\le M_{2} \varepsilon^{\varpi_{2}}, \quad\textit{and} \end{aligned}$$

(4.35)

$$\begin{aligned} &\lim_{\epsilon\to0}\operatorname{dist}_{{\mathcal {H}}_{1,0}}({ \mathcal {E}},{\mathcal {E}}_{\epsilon})=0, \end{aligned}$$

(4.36)

where ${\mathcal {E}}= {\mathcal {L}}{\mathcal {E}}_{0}=\{(\varphi,{\mathscr {L}(\varphi,\vartheta)},\vartheta), (\varphi,\vartheta)\in {\mathcal {E}}_{0}\}$.

Proof

On account of Theorem 3.1, we let $E_{\epsilon}={\mathcal {H}}_{1,\epsilon}$, $V_{\epsilon}= {\mathcal {H}}_{2,\epsilon}$, $W_{\epsilon}= {\mathcal {H}}_{3,\epsilon}$, ${ B}_{\epsilon}= \widetilde{\mathcal {B}}_{4}$ and we check all Assumptions 1–5. To verify Assumption 1, using the interpolation inequality, there exists a constant c such that for some $\theta\in[0,1]$ we have

$$\begin{aligned} \bigl\Vert {\mathscr {L}}(\varphi,\vartheta)-{\mathscr {L}}(\psi,v) \bigr\Vert &\leq \bigl\Vert \Delta(\varphi-\psi) \bigr\Vert + \Vert \varphi-\psi \Vert + \bigl\Vert g(\varphi)-g(\psi ) \bigr\Vert + \Vert \vartheta-v \Vert \\ &\leq c \bigl( \Vert \varphi-\psi \Vert ^{1/2}+ \Vert \varphi- \psi \Vert _{3}^{1/2} \bigr) \Vert \varphi-\psi \Vert _{1}^{1/2}+ \Vert \vartheta-v \Vert \\ &\le c\bigl( \Vert \varphi-\psi \Vert _{1}^{1/2}+ \Vert \vartheta-v \Vert ^{1/2}\bigr), \end{aligned}$$

(4.37)

for any $(\varphi,\vartheta)$ and $(\psi,v)$ in ${\mathcal {B}}$.

Assumptions 2, 4 and 5 were checked in Theorem 3.2. Assumption 3 follows from (4.14) and (4.15). This shows the existence of exponential attractors in ${\mathcal {H}}_{1,0}$ that satisfy (4.34), (4.35) and (4.36). □

We also state the following theorem, which is a direct consequence of Corollary 4.33.

Theorem 4.2

For every $\epsilon\in(0,1]$, there exists a constant $M_{1}>0$ independent of ϵ such that the family of exponential attractors ${\mathcal {E}}_{\epsilon}$ of the semigroup $S_{\epsilon}(t)$ on ${\mathcal {H}}_{1,\epsilon}$ satisfies

$$\begin{aligned} \operatorname{dist}^{\mathrm{sym}}_{{\mathcal {H}}_{1,0}}( \Pi_{\epsilon}{\mathcal {E}}_{\epsilon}, {\mathcal {E}}_{0})\leq M_{1}\sqrt[4]{\epsilon}. \end{aligned}$$

(4.38)

5 Conclusion

In this work, we considered a parabolic–hyperbolic phase-field system, a model which describes phase separation in material sciences. An example is melting and solidification processes. We constructed a robust family of exponential attractors, which are both upper and lower semicontinuous at the parameter $\epsilon=0$. A consequence of this is the existence of fractal dimensional global attractor and, moreover, the dynamics of the global attractor converges to that of the well known Cagilnap phase-field system. Most interestingly, estimates were obtained in norms which are independent of the parameter ϵ.