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

TL;DR

This work analyzes the blow-up behavior of the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in potentially non-convex and unbounded domains. The authors develop a quasi-monotonicity formula for the Giga--Kohn energy in backward similarity variables, accommodating boundary contributions that arise in non-convex geometries and allowing sign-changing solutions. A bootstrap scheme, combined with interpolation and maximal parabolic regularity and localization via cutoff functions, yields uniform $L^{\infty}_t L^q_x$ control for all $p>1$ and $q<p+1$, which in turn implies the type I blow-up rate when $p<p_S=\frac{n+2}{n-2}$ (or infinity in low dimensions). Consequently, in the energy-subcritical range $(n-2)p<n+2$, the paper proves nonexistence of type II blow-up and deduces blow-up of the scaling-critical norm with $q_c=\frac{n(p-1)}{2}$, resolving a decades-old question and extending the type I blow-up theory to general domains and sign-changing data. The results rely on a detailed energy analysis in backward variables, boundary term estimates, and localization techniques that together bypass convexity and positivity assumptions.

Abstract

We consider the semilinear heat equation $u_t=Δu+|u|^{p-1} u$ 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.

Theorems & Definitions (34)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Proposition \ref{['pro:qmono']}