Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation
Noriko Mizoguchi, Philippe Souplet
TL;DR
The paper resolves whether the critical $L^{q^*}$ norm must blow up at finite-time blow-up for the semilinear heat equation, showing that type I blow-up necessarily drives $\|u(t)\|_{L^{q^*}}$ to infinity globally and locally, independent of domain or sign. The authors develop a framework using backward similarity variables, rescaling, and delayed smoothing in weighted spaces, together with backward uniqueness and unique continuation, to rule out the possibility of bounded critical norms under type I blow-up. A notable corollary is the nonexistence of nontrivial self-similar profiles in $L^{q^*}$, while Appendix confirms sharpness by exhibiting type II blow-up with bounded $L^{q^*}$ norms in special cases ($p=p_S$, $N=4$ or $5$). The work thus clarifies the relationship between blow-up rates and critical-norm behavior and provides new insight into concentration phenomena near blow-up points with potential implications for broader nonlinear parabolic systems.
Abstract
We shed light on a long-standing open question for the semilinear heat equation $u_t = Δu + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$Ω$, 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.