Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption.

*(English)*Zbl 0799.35111Summary: This paper deals with the Cauchy problem

$${u}_{t}-{u}_{xx}+{u}^{p}=0;\phantom{\rule{1.em}{0ex}}-\infty <x<+\infty ,\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{2.em}{0ex}}u(x,0)={u}_{0}\left(x\right),$$

where $0<p<1$ and ${u}_{0}\left(x\right)$ is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time $T$, and interfaces separating the regions where $u(x,t)>0$ and $u(x,t)=0$ appear when $t$ is close to $T$. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.