Blow-up and global asymptotics of the limit unstable Cahn--Hilliard equation

Abstract

We study the asymptotic behavior of classes of global and blow-up solutions of a semilinear parabolic equation of the "limit" Cahn--Hilliard type \[u_t = -\Delta(\Delta u + |u|^{p-1}u)\quad \mbox{in} \,\,\, \ren \times \re_+, \quad p>1, \] with bounded integrable initial data. We show that in some $\{p,N\}$-parameter ranges it admits a {\em countable} set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exponent $p=p_0=1+\frac 2N$, where the first simplest profile is shown to be stable. Unlike the blow-up case, we show that, for $p=p_0$, the set of global decaying source-type similarity solutions is {\em continuous} and determine the stable mass-branch. We prove that there exists a countable spectrum of critical exponents $\{p=p_l=1+\frac 2{N+l}, \,\, l =0,1,2,\ldots\}$ creating bifurcation branches, which play a key role in general description of solutions globally decaying as $t \to \infty$.