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

Abstract

We consider $u(x,t)$, a solution of ${\partial }_{t}u=\Delta u+|u{|}^{p-1}u$ 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 $S$.

arXiv comment27 pages