Blow-up behaviour of one-dimensional semilinear parabolic equations

Abstract

Consider the Cauchy problem

where $u_0(x)$ is continuous, nonnegative and bounded, and $F(u)=u^p$ with $p>1$, or $F(u)=e^u$. Assume that $u$ blows up at $x=0$ and $t=T>0$. In this paper we shall describe the various possible asymptotic behaviours of $u(x,t)$ as $(x,t)\to (0,T)$. Moreover, we shall show that if $u_0(x)$ has a single maximum at $x=0$ and is symmetric, $u_0(x)=u_0(-x)$ for $x>0$, there holds

1) If $\mathrm{F}(u)=u^p$ with $p>1$, then

uniformly on compact sets $|\xi |\leqq R$ with $R>0$,

2) If $F(u)=e^u$, then

uniformly on compact sets $|\xi |\leqq R$ with $R>0$.

Résumé

On considère le problème de Cauchy

où $u_0(x)$ est une fonction continue, non négative et bornée, et $F(u)=u^p$ avec $p>1$ ou $F(u)=e^u$. Nous supposons que $u$ explose au point $x=0$ en temps $T>0$. Dans ce travail, nous obtenons tous les comportements asymptotiques possibles de la solution $u(x,t)$ quand $(x,t)\to (0,T)$.