Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation

TL;DR

This paper establishes that under certain conditions, the blow-up set of solutions to a semilinear heat equation is a smooth manifold, by deriving a refined asymptotic behavior that improves understanding of the blow-up profile.

Contribution

It introduces a novel approach to analyze blow-up behavior using non-explicit asymptotics, leading to regularity results for the blow-up set.

Findings

Blow-up set containing a continuum of dimension (N−ℓ) is a C^2 manifold.

Refined asymptotics avoid logarithmic scales, reaching polynomial small terms.

Geometric constraints on the blow-up set improve its regularity.

Abstract

We consider $u(x,t)$, a solution of $\partial_tu = \Delta u + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to \mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensional continuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a $\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined behavior, we have to abandon the explicit profile function as a first order approximation and take a non-explicit function as a first order description of the singular behavior. This way we escape logarithmic scales of the variable $(T-t)$ and reach significant small terms in the polynomial order $(T-t)^\mu$ for some $\mu > 0$. The refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on $S$.