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

TL;DR

This paper investigates the behavior of solutions near non-isolated blow-up points for a semilinear heat equation in two dimensions, revealing specific rigidity and asymptotic properties depending on the degeneracy of the blow-up.

Contribution

It characterizes the asymptotic behavior of solutions near non-isolated blow-up points with degenerate Taylor expansions, introducing new conditions on the approach of blow-up points.

Findings

Either the blow-up points approach with a quadratic relation or a logarithmic-modulated relation.

The parameters  and  are rational with  in (0,2], and the limit L solves a polynomial.

Special case m(a)=4 yields specific  and  values, =0, =3/2 or 2.

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 $\beta =2$.