Threshold, subthreshold and global unbounded solutions of superlinear heat equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909870378188800 |
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| author | Quittner, Pavol Souplet, Philippe |
| author_facet | Quittner, Pavol Souplet, Philippe |
| contents | We consider the semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions, lying on or below the boundary between blow-up and global existence, respectively. For the Cauchy-Dirichlet problem, we prove the boundedness and decay to zero of any subthreshold solution. This implies, in particular, that all global unbounded solutions -- if they exist -- are threshold solutions. For the Cauchy problem, these properties fail in general but we show that they become true for a suitably modified notion of threshold. Our results strongly improve known results even in the model case of power nonlinearities, especially in the Sobolev critical and supercritical cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Threshold, subthreshold and global unbounded solutions of superlinear heat equations Quittner, Pavol Souplet, Philippe Analysis of PDEs 35K58, 35K57, 35B40, 35B44 We consider the semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions, lying on or below the boundary between blow-up and global existence, respectively. For the Cauchy-Dirichlet problem, we prove the boundedness and decay to zero of any subthreshold solution. This implies, in particular, that all global unbounded solutions -- if they exist -- are threshold solutions. For the Cauchy problem, these properties fail in general but we show that they become true for a suitably modified notion of threshold. Our results strongly improve known results even in the model case of power nonlinearities, especially in the Sobolev critical and supercritical cases. |
| title | Threshold, subthreshold and global unbounded solutions of superlinear heat equations |
| topic | Analysis of PDEs 35K58, 35K57, 35B40, 35B44 |
| url | https://arxiv.org/abs/2504.11813 |