    October  2021, 20(10): 3655-3682. doi: 10.3934/cpaa.2021125

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

 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia

* Corresponding author

Received  December 2020 Revised  June 2021 Published  October 2021 Early access  July 2021

We consider the conserved phase-field system
 $\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$
where
 $\tau>0$
is a relaxation time,
 $\delta>0$
is the viscosity parameter,
 $\epsilon\in (0,1]$
is the heat capacity,
 $\phi$
is the order parameter,
 $u$
is the absolute temperature, the Laplace operator
 $N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega)$
is subject to either Neumann boundary conditions (in which case
 $\Omega\subset{\mathbb R}^d$
is a bounded domain with smooth boundary) or periodic boundary conditions (in which case
 $\Omega = \Pi_{i = 1}^d(0,L_i),$
 $L_i>0$
),
 $d = 1,2$
or 3, and
 $G(\phi) = \int_0^\phi g(\sigma)d\sigma$
is a double-well potential. Let
 $j = 1$
when
 $d = 1$
and
 $j = 2$
when
 $d = 2$
or 3. We assume that
 $g\in{\mathcal C}^{j+1}(\mathbb R)$
and satisfies the conditions
 $g'(\phi)\geq -{\mathscr C}_1$
,
 $G(\phi)\ge -{\mathscr C}_2$
and
 $(\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi))$
(
 ${\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho)$
,
 $l = 3,4$
, whenever
 $|m(\phi)|\le \varrho$
), where
 $\varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0$
,
 ${\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6>0$
and
 $m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx$
. For instance,
 $g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k,$
 $p\in{\mathbb N},$
 $p\ge 2,$
 $a_{2p-1}>0,$
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
 ${\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2})$
equipped with the norm
 $\|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2}$
. Moreover, we demonstrate that the global attractor is upper semicontinuous at
 $\epsilon = 0$
in the metric induced by the norm
 $\|.\|_{{\mathcal V}_{j+1}}$
. In addition, the exponential attractors are proven to be Hölder continuous at
 $\epsilon = 0$
in the metric induced by the norm
 $\|.\|_{{\mathcal V}_{j}}$
. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition
 $|g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right),$
 ${\mathscr C}_7>0$
,
 $p>0$
is arbitrary when
 $d = 1, 2$
and
 $p\in [0,3]$
when
 $d = 3$
, was required, preventing
 $g$
to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.
Citation: Ahmed Bonfoh, Ibrahim A. Suleman. Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3655-3682. doi: 10.3934/cpaa.2021125
##### References:
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##### References:
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