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 ${\partial }_{t}u=\Delta u+{\left|u\right|}^{p-1}u$ which blows up at some time $T>0$ , where $u:{ℝ}^N\times [0,T)\to ℝ$ , $p>1$ and $\left(N-2\right)p<N+2$ . Define $S\subset {ℝ}^N$ to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an $\left(N-\ell \right)$ -dimensional continuum for some $\ell \in \{1,\dots ,N-1\}$ , then S is in fact a ${𝒞}^2$ manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, 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 $\left(T-t\right)$ and reach significant small terms in the polynomial order ${\left(T-t\right)}^{\mu }$ for some $\mu >0$ . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.

Keywords: 35B40; 35K57; : 35K55; : 35K50; Blow-Up Profile; Blow-Up Set; Blow-Up Solution; Regularity; Semilinear Heat Equation.