Self-similar solutions of semilinear heat equations with positive speed
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908605009100800 |
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| author | Choi, Kyeongsu Huang, Jiuzhou |
| author_facet | Choi, Kyeongsu Huang, Jiuzhou |
| contents | We classify the smooth self-similar solutions of the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a positive constant after rescaling at the blow-up point for all $p>1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-similar solutions of semilinear heat equations with positive speed Choi, Kyeongsu Huang, Jiuzhou Analysis of PDEs Differential Geometry We classify the smooth self-similar solutions of the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a positive constant after rescaling at the blow-up point for all $p>1$. |
| title | Self-similar solutions of semilinear heat equations with positive speed |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2403.16641 |