IUMJ

Title: Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation

Authors: Christian Stinner and Michael Winkler

Issue: Volume 57 (2008), Issue 5, 2321-2354

Abstract:

This paper deals with the phenomenon of gradient blow-up of nonnegative classical solutions of the Dirichlet problem for \begin{equation}\label{star}\tag{$*$} u_{t} = u^{p} u_{xx} + \kappa u^{r} u_x^{2} + u^{q} \quad \mbox{in } \Omega \times (0,T) \end{equation} in a bounded interval $\Omega \subset \mathbb{R}$, where $p > 2$, $1 \le q \le p - 1$, $r \ge 1$, $\kappa \ge 0$, and the initial data $u_{0}$ are assumed to belong to $C^{1}(\bar{\Omega})$ and satisfy $u_{0}(x) \ge c_{0} \mathrm{dist}(x, \partial\Omega)$ in $\Omega$ with some $c_{0} > 0$. It is shown that if the gradient term in \eqref{star} is weak enough near $u = 0$, then all bounded solutions undergo an infinite time gradient blow-up, whereas if this term is sufficiently strong, then all solutions blow up in $C^{1}(\bar{\Omega})$ within finite time. Here by \emph{weak} we mean that the parameters satisfy either $r = p - 1$ and $\kappa \le p -2$, or $r > p - 1$, and by \emph{strong} the precise opposite, that is, either $r = p - 1$ and $\kappa > p -2$, or $r < p - 1$.