Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn--Hilliard type

Abstract

Fourth-order semilinear parabolic equations of the Cahn--Hilliard-type (01) u_t + \D^2 u = \g u \pm \D (|u|^{p-1}u) in Ω\times \re_+, are considered in a smooth bounded domain $Ø\subset \ren$ with Navier-type boundary conditions on $\p Ø$, or $Ø= \ren$, where $p>1$ and $\g$ are given real parameters. The sign $``+"$ in the "diffusion term" on the right-hand side means the stable case, while $``-"$ reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for $\g=0$, (02) u_t + \D^2 u= \pm \D(|u|^{p-1}u) \inA. The following three main problems are studied: (i) for the unstable model (01), with the $- \D (|u|^{p-1}u)$, existence and multiplicity of classic steady states in $Ø\subset \ren$ and their global behaviour for large $\g>0$; (ii) for the stable model (02), global existence of smooth solutions $u(x,t)$ in $\ren \times \re_+$ for bounded initial data $u_0(x)$ in the subcritical case $p \le p_{*}= 1 + \frac {4}{(N-2)_+}$; and (iii) for the unstable model (02), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.