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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.06363 |
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| _version_ | 1866915857961058304 |
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| author | Benedikt, Jiří Bobkov, Vladimir Dhara, Raj Narayan Girg, Petr |
| author_facet | Benedikt, Jiří Bobkov, Vladimir Dhara, Raj Narayan Girg, Petr |
| contents | We show that the parabolic equation $u_t + (-Δ)^s u = q(x) |u|^{α-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times (\mathbb{R}^N \setminus Ω)$, where $Ω$ is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided $α\in (0,1)$ and the nonnegative bounded weight function $q$ is separated from zero on an open subset of $Ω$. This fact contrasts with the (super)linear case $α\geq 1$ in which the only bounded finite energy solution is identically zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06363 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity Benedikt, Jiří Bobkov, Vladimir Dhara, Raj Narayan Girg, Petr Analysis of PDEs 35A01, 35A02, 35B30, 35K58, 35R11 We show that the parabolic equation $u_t + (-Δ)^s u = q(x) |u|^{α-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times (\mathbb{R}^N \setminus Ω)$, where $Ω$ is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided $α\in (0,1)$ and the nonnegative bounded weight function $q$ is separated from zero on an open subset of $Ω$. This fact contrasts with the (super)linear case $α\geq 1$ in which the only bounded finite energy solution is identically zero. |
| title | Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity |
| topic | Analysis of PDEs 35A01, 35A02, 35B30, 35K58, 35R11 |
| url | https://arxiv.org/abs/2302.06363 |