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arXiv 2103.12795math.AP
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半线性heat equation非孤立爆破点附近的行为刚度

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

Frank Merle, Hatem Zaag

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摘要：

考虑了具有Sobolev次临界幂非线性项的半线性热方程在$N=2$维上的解，其中$u(x,t)$是在有限时间$T$上爆破的解.给定一个非孤立的爆破点$a$，我们假设解在$(a,T)$附近的泰勒展开服从由某个偶数$m(a)\ge 4$标记的退化情形.如果我们有一个序列$a_n \to a$作为$n\to \infty$，我们表明在坐标改变和提取子序列之后，对于一些$L>0$，%直到提取子序列仍然表示相同的$a_{n,1}-a_1 = o((a_{n,2}-a_2)^2)$或$|a_{n,1}-a_1||a_{n,2}-a_2|^{-β} |\log|a_{n,2}-a_2||^{-α} \to L> 0$，其中，$α$和$β$具有有限数量的有理数值，其中，$β\in(0,2]$和$L$是多项式方程的解，该多项式方程取决于溶液如果是$m(a)=4$，则为$α=0$和$β=3/2$或$β=2$。

英文摘要：

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 $|a_{n,1}-a_1||a_{n,2}-a_2|^{-β} |\log|a_{n,2}-a_2||^{-α} \to L> 0$ for some $L>0$, %up to extracting a subsequence still denoted the same, where $α$ and $β$ enjoy a finite number of rational values with $β\in(0,2]$ 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 $α=0$ and either $β=3/2$ or $β=2$.

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