Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929271477370880 |
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| author | de Pablo, Arturo Quirós, Fernando Ruiz-Cases, Jorge |
| author_facet | de Pablo, Arturo Quirós, Fernando Ruiz-Cases, Jorge |
| contents | We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation,
\[\partial_t u + \mathcal{L}φ(u) = 0,\]
and the interplay between these two properties. Here $\mathcal{L}$ is a stable-like operator and $φ$ is a singular nonlinearity.
We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation.
The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where $φ$ is a power. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation de Pablo, Arturo Quirós, Fernando Ruiz-Cases, Jorge Analysis of PDEs 35R11, 35K67, 35B09, 35B40 We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, \[\partial_t u + \mathcal{L}φ(u) = 0,\] and the interplay between these two properties. Here $\mathcal{L}$ is a stable-like operator and $φ$ is a singular nonlinearity. We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation. The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where $φ$ is a power. |
| title | Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation |
| topic | Analysis of PDEs 35R11, 35K67, 35B09, 35B40 |
| url | https://arxiv.org/abs/2403.06647 |