Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption

Abstract

We investigate the phenomenon of instantaneous shrinking of the support of nonnegative solutions to the Cauchy problem in $\\mathbb R^n$ for $$ u_t=f(u) \Delta u - g(u), \quad \mbox{where } f(0)=0. $$ Among other results, it is shown by means of comparison and integral techniques that under some structural assumptions on $f$ and $g$, a necessary {\em and} sufficient condition on the growth of $g$ near zero for instantaneous shrinking to occur is $\int_0^1 \frac{ds}{g(s)} <\infty$.