The global profile of blow-up at space infinity in semilinear heat equations

Abstract

We consider semilinear heat equations on ${\mathbb{R}}^N$ and discuss the blow-up of solutions that occurs only at space infinity. We give sufficient conditions for such phenomena, and study the global profile of solutions at the blow-up time. Among other things, we establish a nearly optimal upper bound for the blow-up profile, which shows that the profile $u(x,T)$ cannot grow too fast as $|x|\to \infty$. We also prove that such blow-up is always complete.