Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

Abstract

We study initial boundary value problems for the unstable convective Cahn-Hilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is well-known that without the convective term, the solutions of the unstable CH equation ?t u + ? 4xu + ?2x(|u|pu) = 0 may blow up in ?nite time for anyp > 0. In contrast to that, we show that the presence of the convective term u?xuin the Cahn-Hilliard equation prevents blow up at least for 0 < p <49. We alsoshow that the blowing up solutions still exist if p is large enough (p ? 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.