Uniqueness and root-Lipschitz regularity for a degenerate heat equation
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916321325744128 |
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| author | Dunlap, Alexander Graham, Cole |
| author_facet | Dunlap, Alexander Graham, Cole |
| contents | We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that ensures uniqueness. For suitable initial data, we prove a lower bound on the time for which a strong solution $u$ exists and $\sqrt{u}$ remains globally Lipschitz in space. In a companion paper, we show that this condition is important in the study of two-dimensional nonlinear stochastic heat equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11820 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniqueness and root-Lipschitz regularity for a degenerate heat equation Dunlap, Alexander Graham, Cole Analysis of PDEs Probability 35B65 (Primary), 35B53, 35K59, 35B44, 60H10 (Secondary) We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that ensures uniqueness. For suitable initial data, we prove a lower bound on the time for which a strong solution $u$ exists and $\sqrt{u}$ remains globally Lipschitz in space. In a companion paper, we show that this condition is important in the study of two-dimensional nonlinear stochastic heat equations. |
| title | Uniqueness and root-Lipschitz regularity for a degenerate heat equation |
| topic | Analysis of PDEs Probability 35B65 (Primary), 35B53, 35K59, 35B44, 60H10 (Secondary) |
| url | https://arxiv.org/abs/2308.11820 |