Attractors of semigroups associated with nonlinear systems for diffusive phase separation

Abstract

We consider a model for diffusive phase transitions, for instance, the component separation in a binary mixture. Our model is described by two functions, the absolutete temperature $\theta :=\theta \left(t,x\right)$ and the order parameter $w:=w\left(t,x\right)$, which are governed by a system of two nonlinear parabolic PDEs. The order parameter $w$ is constrained to have double obstacles ${\sigma }_{*}\le w\le {\sigma }^{*}$ (i.e., ${\sigma }_{*}$ and ${\sigma }^{*}$ are the threshold values of $w$). The objective of this paper is to discuss the semigroup $\left\{S\left(t\right)\right\}$ associated with the phase separation model, and construct its global attractor.