Behavior Rigidity Near Non-Isolated Blow-up Points for the Semilinear Heat Equation

Abstract

\n We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution that 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 $L>0$, 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 $\\beta =2$.