半线性热方程无穷多个奇异自相似解的存在性
Existence of infinitely many singular self-similar solutions for the semilinear heat equation
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中文总结 AI 辅助
本文证明半线性热方程在特定参数范围内存在无穷多个奇异正自相似解，并研究其渐近行为与轮廓界，进而利用这些解证明柯西问题正解的非唯一性。
中文摘要 AI 辅助
我们证明了半线性热方程 $\partial_t u = \Delta u + u^p$（其中 $x \in \R^N$, $t > 0$, $N \geq 3$ 且 $N/(N-2) \leq p \leq (N+2)/(N-2)$）存在无穷多个奇异正向前自相似解。我们还分别研究了当 $|x| \to 0$ 和 $|x| \to \infty$ 时解的渐近行为和解的轮廓的界。此外，通过使用奇异自相似解，我们证明了当 $p = N/(N-2)$ 时该方程柯西问题正解的非唯一性。
英文摘要
We show the existence of infinitely many singular positive forward self-similar solutions for the semilinear heat equation $\partial_t u = Δu + u^p$ for $x \in \R^N$, $t > 0$, where $N \geq 3$ and $N/(N-2) \leq p \leq (N+2)/(N-2)$. We also investigate the asymptotic behavior and the bound of the profiles of the solutions as $|x| \to 0$ and $|x| \to \infty$, respectively. Furthermore, we show nonuniqueness of positive soutions of the Cauchy problem for the equation with $p = N/(N-2)$ by using singular self-similar solutions.
发表机构
- Department of Mathematics, Hiroshima University（广岛大学数学系）
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