Unstable blow-up patterns

Abstract

Consider the semilinear heat equation $u_{t}=\Delta u + u^{p}$, $t>0$, in an open set $\Omega \subset \mathbf {R}^N$, $p>1$. For every $a \in \Omega$, we construct solutions which blow up at $a$, at a finite time $T$, according to a variety of specific asymptotic behaviors. These blow-up patterns are unstable. The corresponding solutions have an arbitrary large number of local maxima, collapsing at $a$ for $t=T$.