Modulus of continuity for solutions of non-local heat equations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915614374756352 |
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| author | Andrews, Ben Chen, Sophie |
| author_facet | Andrews, Ben Chen, Sophie |
| contents | We extend the method of modulus of continuity for solutions of parabolic equations--as used, for instance, to prove the Fundamental Gap Conjecture--to solutions of non-local heat equations on R^n and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modulus of continuity for solutions of non-local heat equations Andrews, Ben Chen, Sophie Analysis of PDEs We extend the method of modulus of continuity for solutions of parabolic equations--as used, for instance, to prove the Fundamental Gap Conjecture--to solutions of non-local heat equations on R^n and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain. |
| title | Modulus of continuity for solutions of non-local heat equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.19023 |