Behavior rigidity near non-isolated blow-up points for the semilinear heat equation

Abstract

We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution which blows up in finite time $T$. Given a non isolated blow-up point $a$, we assume that the Taylor expansion of the solution near $(a,T)$ obeys some degenerate situation labeled by some even integer $m(a)\ge 4$. If we have a sequence $a_n \to a$ as $n\to \infty$, we show after a change of coordinates and the extraction of a subsequence that either $a_{n,1}-a_1 = o((a_{n,2}-a_2)^2)$ or $|a_{n,1}-a_1||a_{n,2}-a_2|^{-\beta} |\log|a_{n,2}-a_2||^{-\alpha} \to L> 0$ for some $L>0$, %up to extracting a subsequence still denoted the same, where $\alpha$ and $\beta$ enjoy a finite number of rational values with $\beta \in(0,2]$ and $L$ is a solution of a polynomial equation depending on the coefficients of the Taylor expansion of the solution. If $m(a)=4$, then $\alpha=0$ and either $\beta=3/2$ or $\beta =2$.