Global behavior of nonlocal in time reaction-diffusion equations
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915121210589184 |
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| author | Torebek, Berikbol T. |
| author_facet | Torebek, Berikbol T. |
| contents | The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation $$\partial_t (k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\inΩ\subset\mathbb{R}^n, t>0,$$ where $k\in L^1_{loc}(\mathbb{R}_+),$ $f$ is a locally Lipschitz function, $\mathcal{L}_x$ is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08985 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global behavior of nonlocal in time reaction-diffusion equations Torebek, Berikbol T. Analysis of PDEs Primary 35K55, 35R11, Secondary 35K57, 35A01 The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation $$\partial_t (k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\inΩ\subset\mathbb{R}^n, t>0,$$ where $k\in L^1_{loc}(\mathbb{R}_+),$ $f$ is a locally Lipschitz function, $\mathcal{L}_x$ is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented. |
| title | Global behavior of nonlocal in time reaction-diffusion equations |
| topic | Analysis of PDEs Primary 35K55, 35R11, Secondary 35K57, 35A01 |
| url | https://arxiv.org/abs/2310.08985 |