Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Abstract

We consider solutions of quasilinear equations $u_t=\Delta u^m + u^p$ in $\mathbb R^N$ with the initial data $u_0$ satisfying $0 < u_0< M$ and $\lim_{|x|\to\infty}u_0(x)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout $\mathbb R^N$ when $m>p>1$.

Keywords

• quasilinear heat equation
• total blow-up
• blow-up only at space infinity