Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry

Baiyu Liu; Wenlong Yang; Baiyu Liu; Wenlong Yang

doi:10.3934/cam.2025014

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 341-364. doi: 10.3934/cam.2025014

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Research article

Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry

Baiyu Liu ^,,
Wenlong Yang

School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District Beijing, 100083, P.R. China

Received: 15 September 2024 Revised: 12 March 2025 Accepted: 18 March 2025 Published: 09 April 2025
35R11, 35B07

In this paper, we investigated the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. First, we discussed the properties of solutions in the elliptic space. Then, we established crucial principles, including the asymptotic narrow region principle. Finally, we employed the method of moving planes to demonstrate the asymptotic symmetry of the solutions.
- parabolic equation,
- asymptotic symmetry,
- monotonicity,
- elliptic geometry
Citation: Baiyu Liu, Wenlong Yang. Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry[J]. Communications in Analysis and Mechanics, 2025, 17(2): 341-364. doi: 10.3934/cam.2025014

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Abstract

In this paper, we investigated the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. First, we discussed the properties of solutions in the elliptic space. Then, we established crucial principles, including the asymptotic narrow region principle. Finally, we employed the method of moving planes to demonstrate the asymptotic symmetry of the solutions.

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