IUMJ

Title: Blow-up in a degenerate parabolic equation

Authors: Michael Winkler

Issue: Volume 53 (2004), Issue 5, 1415-1442

Abstract:

This paper investigates the asymptotic behavior of positive blow-up solutions to the Dirichlet problem for $u_t=u^p(\Delta u+u)$, $p \ge 2$, in bounded domains $\Omega \subset \mathbb{R}^n$. It is proved that for any initial datum, blow-up occurs essentially faster than at the ODE rate $(T-t)^{-1/p}$, and that the subset of $\bar{\Omega}$ of points of such fast blow-up has positive measure. In the one-dimensional case, asymptotic profiles of solutions are discussed. Concerning the corresponding Cauchy problem, which has explicit spatially homogeneous solutions blowing up exactly at the ODE rate, it is shown that even arbitrarily slow spatial decay of the initial datum can enforce fast blow-up.