Blow-up of the critical norm for a supercritical semilinear heat equation

TL;DR

This paper proves that for supercritical semilinear heat equations, the critical Lebesgue norm must diverge near blow-up, except at the critical exponent where bounded solutions exist, highlighting the nature of blow-up behavior.

Contribution

It establishes the unboundedness of the critical norm for supercritical blow-up solutions without type I assumptions, clarifying blow-up dynamics in this regime.

Findings

Critical norm diverges near blow-up for p > p_S.

Existence of bounded critical norm solutions at p = p_S.

Optimality of the supercritical range for blow-up behavior.

Abstract

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range $p>p_S$ is optimal in view of the existence of type II blow-up solutions with bounded critical norm for $p=p_S$.