Nonnegative solutions to nonlocal parabolic equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916735121096704 |
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| author | Liao, Naian Weidner, Marvin |
| author_facet | Liao, Naian Weidner, Marvin |
| contents | We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain translation invariant operators, and new ideas are needed in our general setting. Second, we establish sharp two-sided bounds for the fundamental solution via purely variational techniques, entirely bypassing tools from semigroup theory, Dirichlet forms, and stochastic analysis. Third, we derive sharp Harnack-type estimates that are novel even for the fractional heat equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonnegative solutions to nonlocal parabolic equations Liao, Naian Weidner, Marvin Analysis of PDEs Probability 47G20, 35B65, 35R09, 31B05, 35C15, 35K08, 35A02 We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a result has previously been studied only for certain translation invariant operators, and new ideas are needed in our general setting. Second, we establish sharp two-sided bounds for the fundamental solution via purely variational techniques, entirely bypassing tools from semigroup theory, Dirichlet forms, and stochastic analysis. Third, we derive sharp Harnack-type estimates that are novel even for the fractional heat equation. |
| title | Nonnegative solutions to nonlocal parabolic equations |
| topic | Analysis of PDEs Probability 47G20, 35B65, 35R09, 31B05, 35C15, 35K08, 35A02 |
| url | https://arxiv.org/abs/2505.08449 |