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 $\beta =2$.

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