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+∂4x+∂2x(|u|pu)=0 may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 0 < p < < 4/9. We also show 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. © 2013 American Institute of Physics.