Liouville Theorems for Fractional Parabolic Equations
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains,...
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| Format: | Article |
| Language: | English |
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De Gruyter
2021-11-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2021-2148 |
| _version_ | 1811185921344667648 |
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| author | Chen Wenxiong Wu Leyun |
| author_facet | Chen Wenxiong Wu Leyun |
| author_sort | Chen Wenxiong |
| collection | DOAJ |
| description | In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u→0u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space Rn-1×R\mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems. |
| first_indexed | 2024-04-11T13:37:36Z |
| format | Article |
| id | doaj.art-65b486acdae14e98a51f75af5252c5b6 |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-11T13:37:36Z |
| publishDate | 2021-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-65b486acdae14e98a51f75af5252c5b62022-12-22T04:21:24ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752021-11-0121493995810.1515/ans-2021-2148Liouville Theorems for Fractional Parabolic EquationsChen Wenxiong0Wu Leyun1Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, USASchool of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, P. R. China; and Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong KongIn this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u→0u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space Rn-1×R\mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.https://doi.org/10.1515/ans-2021-2148liouville type theoremsfractional parabolic equationsentire solutionsmonotonicitynonexistence of solutionsnarrow region principlesmaximum principle for antisymmetric functions35b5335r1135k58 |
| spellingShingle | Chen Wenxiong Wu Leyun Liouville Theorems for Fractional Parabolic Equations Advanced Nonlinear Studies liouville type theorems fractional parabolic equations entire solutions monotonicity nonexistence of solutions narrow region principles maximum principle for antisymmetric functions 35b53 35r11 35k58 |
| title | Liouville Theorems for Fractional Parabolic Equations |
| title_full | Liouville Theorems for Fractional Parabolic Equations |
| title_fullStr | Liouville Theorems for Fractional Parabolic Equations |
| title_full_unstemmed | Liouville Theorems for Fractional Parabolic Equations |
| title_short | Liouville Theorems for Fractional Parabolic Equations |
| title_sort | liouville theorems for fractional parabolic equations |
| topic | liouville type theorems fractional parabolic equations entire solutions monotonicity nonexistence of solutions narrow region principles maximum principle for antisymmetric functions 35b53 35r11 35k58 |
| url | https://doi.org/10.1515/ans-2021-2148 |
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