Optimal condition for blow-up of the critical Lq norm for the semilinear heat equation

Abstract

We shed light on a long-standing open question for the semilinear heat equation ut=Δu+|u|p−1u. Namely, without any restriction on the exponent p>1 nor on the smooth domain Ω, we prove that the critical Lq norm blows up whenever the solution undergoes type I blow-up. A similar property is also obtained for the local critical Lq norm near any blow-up point.In view of recent results of existence of type II blow-up solutions with bounded critical Lq 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 Lq 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 backward uniqueness and unique continuation properties for parabolic equations.As a by-product, we obtain the nonexistence of self-similar profiles in the critical Lq space. Such properties were up to now only known for p≤pS and in radially symmetric case for p>pS, where pS is the Sobolev exponent.