Asymptotic profile of solutions for the limit unstable Cahn-Hilliard equation with inertial term

Abstract

The aim of this paper is to investigate the asymptotic behavior of solutions to the limit unstable Cahn-Hilliard equation with inertial term $$ \partial_t^2 u + \partial_t u + \Delta ( \Delta u + | u |^{p-1} u)=0, \quad {\rm in } \quad (0,\infty) \times {\mathbb{R}}^n. $$ We shall prove the unique global existence of a small solution for the equation in the super critical case $p > 1 + \frac{2}{n}$ in $n=1,2,3$, and also give an asymptotic profile of the solution. By considering the higher-order expansion of the solution, we obtain more precise information about optimal decay of the solution under the more restricted condition $p > 1+ \frac{4}{n}$, and observe the contribution of the nonlinear term to the solution.