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 $\beta =2$.

中文翻译：

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半线性热方程的非孤立爆破点附近的行为刚度

我们考虑在维度 $N=2$ 中具有 Sobolev 次临界功率非线性的半线性热方程，并且 $u(x,t)$ 是在有限时间 $T$ 内爆炸的解。给定一个非孤立的爆破点$a$，我们假设$(a,T)$ 附近解的泰勒展开服从某些偶数$m(a)\ge 4$ 标记的退化情况。如果我们有一个序列 $a_n \to a$ as $n\to \infty $，我们会在坐标改变和提取子序列后显示 ${a_{n,1}}-a_1 = o(( a_{n,2}-a_2)^2)$ 或 $|a_{n,1}-a_1||a_{n,2}-a_2|^{-\beta } |\log |a_{n,2 }-a_2||^{-\alpha } \to L> 0$ 对于某些 $L>0$，其中 $\alpha $ 和 $\beta $ 享有有限数量的有理值，其中 $\beta \in (0,2]$ 和 $L$ 是多项式方程的解取决于解的泰勒展开系数. 如果 $m(a)=4$,