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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.14428 |
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| _version_ | 1866916295923990528 |
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| author | Ferreira, Raúl de Pablo, Arturo |
| author_facet | Ferreira, Raúl de Pablo, Arturo |
| contents | We study the blow-up question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial Lévy kernel. We show that the existence of solutions that blow up in finite time or exist globally depends only on the behaviour of the spatial kernel at infinity. A main difficulty of the work stems from estimating the fundamental pair defining the solution through a Duhamel formula, due to the generality of the setting, which includes singular or not, at the origin, spatial kernels, that can be either positive or compactly supported.
As a byproduct we obtain that the Fujita exponent for the fractional type operators similar to the Caputo fractional derivative and the fractional Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up for a double nonlocal heat equation Ferreira, Raúl de Pablo, Arturo Analysis of PDEs 35C15, 35K57, 35B33, 35B44, We study the blow-up question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial Lévy kernel. We show that the existence of solutions that blow up in finite time or exist globally depends only on the behaviour of the spatial kernel at infinity. A main difficulty of the work stems from estimating the fundamental pair defining the solution through a Duhamel formula, due to the generality of the setting, which includes singular or not, at the origin, spatial kernels, that can be either positive or compactly supported. As a byproduct we obtain that the Fujita exponent for the fractional type operators similar to the Caputo fractional derivative and the fractional Laplacian. |
| title | Blow-up for a double nonlocal heat equation |
| topic | Analysis of PDEs 35C15, 35K57, 35B33, 35B44, |
| url | https://arxiv.org/abs/2406.14428 |