The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929740500172800 |
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| author | Miyamoto, Reiri Sobajima, Motohiro |
| author_facet | Miyamoto, Reiri Sobajima, Motohiro |
| contents | This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation}
\begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\ u(x,0)=u_0(x)\ge0,&x\in\mathbb{R},
\end{cases} \end{equation} where $p>1$, $m\ge0$, $u_0\in BC(\mathbb{R})$ and the potential $V=V(x)\in BC(\mathbb{R})$ satisfies a certain property. More precisely, we determine the critical Fujita exponent for (P), that is, the threshold for the global existence/nonexistence of (P). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities Miyamoto, Reiri Sobajima, Motohiro Analysis of PDEs 35K58, 35B33 This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\ u(x,0)=u_0(x)\ge0,&x\in\mathbb{R}, \end{cases} \end{equation} where $p>1$, $m\ge0$, $u_0\in BC(\mathbb{R})$ and the potential $V=V(x)\in BC(\mathbb{R})$ satisfies a certain property. More precisely, we determine the critical Fujita exponent for (P), that is, the threshold for the global existence/nonexistence of (P). |
| title | The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities |
| topic | Analysis of PDEs 35K58, 35B33 |
| url | https://arxiv.org/abs/2503.02446 |