On a generalized Cahn--Hilliard model with $p$-Laplacian

Abstract

A generalized Cahn-Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label"generalized"refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p>1$ and a double well potential of the form $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$, with $\theta>1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F"(\pm1)>0$, which are typical of the standard Cahn-Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $\theta\geq p>1$. In the $critical$ $\theta=p>1$, we prove $exponentially$ $slow$ $motion$ of profiles with a transition layer structure, thus extending the well know results of the standard model, where $\theta=p=2$; conversely, in the $supercritical$ case $\theta>p>1$, we prove $algebraic$ $slow$ $motion$ of layered profiles.