Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term

We consider the conserved phase-field system\begin{document}$\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$\end{document}where \begin{document}$ \tau>0 $\end{document} is a relaxation time, \begin{document}$ \delta>0 $\end{document} is the viscosity parameter, \begin{document}$ \epsilon\in (0,1] $\end{document} is the heat capacity, \begin{document}$ \phi $\end{document} is the order parameter, \begin{document}$ u $\end{document} is the absolute temperature, the Laplace operator \begin{document}$ N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega) $\end{document} is subject to either Neumann boundary conditions (in which case \begin{document}$ \Omega\subset{\mathbb R}^d $\end{document} is a bounded domain with smooth boundary) or periodic boundary conditions (in which case \begin{document}$ \Omega = \Pi_{i = 1}^d(0,L_i), $\end{document}\begin{document}$ L_i>0 $\end{document} ), \begin{document}$ d = 1,2 $\end{document} or 3, and \begin{document}$ G(\phi) = \int_0^\phi g(\sigma)d\sigma $\end{document} is a double-well potential. Let \begin{document}$ j = 1 $\end{document} when \begin{document}$ d = 1 $\end{document} and \begin{document}$ j = 2 $\end{document} when \begin{document}$ d = 2 $\end{document} or 3. We assume that \begin{document}$ g\in{\mathcal C}^{j+1}(\mathbb R) $\end{document} and satisfies the conditions \begin{document}$ g'(\phi)\geq -{\mathscr C}_1 $\end{document} , \begin{document}$ G(\phi)\ge -{\mathscr C}_2 $\end{document} and \begin{document}$ (\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi)) $\end{document} ( \begin{document}$ {\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho) $\end{document} , \begin{document}$ l = 3,4 $\end{document} , whenever \begin{document}$ |m(\phi)|\le \varrho $\end{document} ), where \begin{document}$ \varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0 $\end{document} , \begin{document}$ {\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6>0 $\end{document} and \begin{document}$ m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx $\end{document} . For instance, \begin{document}$ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $\end{document}\begin{document}$ p\in{\mathbb N}, $\end{document}\begin{document}$ p\ge 2, $\end{document}\begin{document}$ a_{2p-1}>0, $\end{document} satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space \begin{document}$ {\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2}) $\end{document} equipped with the norm \begin{document}$ \|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2} $\end{document} . Moreover, we demonstrate that the global attractor is upper semicontinuous at \begin{document}$ \epsilon = 0 $\end{document} in the metric induced by the norm \begin{document}$ \|.\|_{{\mathcal V}_{j+1}} $\end{document} . In addition, the exponential attractors are proven to be Hölder continuous at \begin{document}$ \epsilon = 0 $\end{document} in the metric induced by the norm \begin{document}$ \|.\|_{{\mathcal V}_{j}} $\end{document} . Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition \begin{document}$ |g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right), $\end{document}\begin{document}$ {\mathscr C}_7>0 $\end{document} , \begin{document}$ p>0 $\end{document} is arbitrary when \begin{document}$ d = 1, 2 $\end{document} and \begin{document}$ p\in [0,3] $\end{document} when \begin{document}$ d = 3 $\end{document} , was required, preventing \begin{document}$ g $\end{document} to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.