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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| Published in | Advanced nonlinear studies Vol. 21; no. 4; pp. 939 - 958 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
De Gruyter
01.11.2021
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1536-1365 2169-0375 |
| DOI | 10.1515/ans-2021-2148 |
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| Abstract | 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
at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space
and obtain some new connections between the nonexistence of solutions in a half space
and in the whole space
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. |
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| AbstractList | 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. 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 → 0 u\to 0 at infinity to a polynomial growth on 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 R n - 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. 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 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space and obtain some new connections between the nonexistence of solutions in a half space and in the whole space 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. |
| Author | Wu, Leyun Chen, Wenxiong |
| Author_xml | – sequence: 1 givenname: Wenxiong surname: Chen fullname: Chen, Wenxiong email: wchen@yu.edu organization: Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, USA – sequence: 2 givenname: Leyun surname: Wu fullname: Wu, Leyun email: leyunwu@sjtu.edu.cn organization: School 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 Kong |
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| Snippet | In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed... |
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| SubjectTerms | 35B53 35K58 35R11 Entire Solutions Fractional Parabolic Equations Liouville Type Theorems Maximum Principle for Antisymmetric Functions Monotonicity Narrow Region Principles Nonexistence of Solutions |
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| Title | Liouville Theorems for Fractional Parabolic Equations |
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