Blow-up rate for the subcritical semilinear heat equation in non-convex domains

TL;DR

This paper proves that solutions to the subcritical semilinear heat equation in non-convex domains cannot blow up in a specific non-standard way, resolving a long-standing open problem from the 1980s.

Contribution

It establishes the nonexistence of type II blow-up for sign-changing solutions in non-convex, possibly unbounded domains within the energy subcritical range.

Findings

No type II blow-up occurs in the specified setting

The blow-up of the scaling critical norm is demonstrated

Results resolve a long-standing open question from the 1980s

Abstract

We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1} u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range $(n-2)p<n+2$. This resolves a long-standing open question dating back to the 1980s and also deduces the blow-up of the scaling critical norm.