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

TL;DR

This paper investigates the effects of convection on the blow-up behavior of solutions to the convective Cahn-Hilliard equation, showing that convection can prevent blow-up for certain nonlinearities but not for larger ones.

Contribution

It demonstrates that the convective term inhibits blow-up for $0<p<rac{4}{9}$ and identifies conditions under which blow-up still occurs for larger $p$, extending understanding of solution behavior.

Findings

Convection prevents blow-up for $0<p<rac{4}{9}$.

Blow-up solutions exist for $p \,\geq 2$.

Analysis extends to related higher-order equations.

Abstract

We study initial boundary value problems for the convective Cahn-Hilliard equation $\Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0$. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term $u\px u$ in the Cahn-Hilliard equation prevents blow up at least for $0<p<\frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($p\ge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.