Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation

Abstract

We shed light on a long-standing open question for the semilinear heat equation $u_t=\Delta u+|u{|}^{p-1}u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$\Omega$, we prove that the critical $L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.} A~similar property is also obtained for the local critical $L^q$ norm near any blow-up point. In view of recent results of existence of type~II blow-up solutions with bounded critical $L^q$ norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical $L^q$ norm blow-up appears to be a completely new observation. Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with {\it backward uniqueness and unique continuation properties} for parabolic equations. As a by-product, we obtain the nonexistence of self-similar profiles in the critical $L^q$ space. Such properties were up to now only known for $p\le p_S$ and in radially symmetric case for $p>p_S$, where $p_S$ is the Sobolev exponent.

Advances in Mathematics, to appear. 16 pages, minor corrections with respect to v1