On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910904495374336 |
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| author | Gonzálvez, Irene Quirós, Fernando Soria, Fernando Vondraček, Zoran |
| author_facet | Gonzálvez, Irene Quirós, Fernando Soria, Fernando Vondraček, Zoran |
| contents | We develop a Widder-type theory for nonlocal heat equations involving quite general Lévy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal Lévy operators defined through positive symmetric Lévy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions Gonzálvez, Irene Quirós, Fernando Soria, Fernando Vondraček, Zoran Analysis of PDEs 35A02, 35C15, 35K08, 35A01, 35S05, 60J60 We develop a Widder-type theory for nonlocal heat equations involving quite general Lévy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal Lévy operators defined through positive symmetric Lévy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part. |
| title | On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions |
| topic | Analysis of PDEs 35A02, 35C15, 35K08, 35A01, 35S05, 60J60 |
| url | https://arxiv.org/abs/2504.04246 |