Liouville theorems and Harnack inequalities for Allen-Cahn type equation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914748531998720 |
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| author | Lu, Zhihao |
| author_facet | Lu, Zhihao |
| contents | We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-Émery curvature condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10760 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Liouville theorems and Harnack inequalities for Allen-Cahn type equation Lu, Zhihao Analysis of PDEs We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-Émery curvature condition. |
| title | Liouville theorems and Harnack inequalities for Allen-Cahn type equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2308.10760 |