Threshold property of a singular stationary solution for semilinear heat equations with exponential growth
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arXiv
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| Natura: | Preprint |
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2024
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| author | Hisa, Kotaro Miyamoto, Yasuhito |
| author_facet | Hisa, Kotaro Miyamoto, Yasuhito |
| contents | Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ such that $u^*(x)\to\infty$ as $|x|\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ and $u_0\not\equiv u^*$, while the problem has no nonnegative local-in-time solutions $u$ such that $u\ge u^*$ if $u_0\ge u^*$ and $u_0\not\equiv u^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Threshold property of a singular stationary solution for semilinear heat equations with exponential growth Hisa, Kotaro Miyamoto, Yasuhito Analysis of PDEs 35K15, 35A01, 35A21, 35B44 Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ such that $u^*(x)\to\infty$ as $|x|\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ and $u_0\not\equiv u^*$, while the problem has no nonnegative local-in-time solutions $u$ such that $u\ge u^*$ if $u_0\ge u^*$ and $u_0\not\equiv u^*$. |
| title | Threshold property of a singular stationary solution for semilinear heat equations with exponential growth |
| topic | Analysis of PDEs 35K15, 35A01, 35A21, 35B44 |
| url | https://arxiv.org/abs/2409.16549 |