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Monotonicity and one-dimensional symmetry of solutions for fractional reaction-diffusion equations and various applications of sliding methods

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Abstract

In this paper, we obtain monotonicity and one-dimensional symmetry of entire solutions to fractional reaction-diffusion equations by introducing a sliding method, and thus prove the Gibbons conjecture for entire solutions to fractional parabolic equations. After establishing key ingredients such as a generalized weighted average inequality and maximum principles in unbounded domains, we demonstrate how these new ideas and tools can be employed in carrying out the sliding method to derive monotonicity and one-dimensional symmetry of entire solutions to fractional parabolic equations. We also compare the sliding method with the method of moving planes and list several other interesting applications of the former such as deriving a uniform lower bound for solutions in unbounded domains by sliding centers of balls along a desired path and proving the non-existence of solutions for certain fractional inequalities by sliding the graphs of functions up and down. We believe that these new ideas and methods introduced here can be employed to study many other nonlocal equations, both elliptic and parabolic, with wider range of fractional operators and more general nonlinearities.

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Acknowledgements

We are grateful to the referee for his/her valuable comments, which help us considerably improve the precision and exposition of this paper. The research of Wenxiong Chen was partially supported by Simons Foundation 847690 and NSFC Grants 12071229. The research of Leyun Wu was partially supported by NSFC Grants 12031012.

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  1. Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, USA

    Wenxiong Chen

  2. Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, People’s Republic of China

    Leyun Wu

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Chen, W., Wu, L. Monotonicity and one-dimensional symmetry of solutions for fractional reaction-diffusion equations and various applications of sliding methods. Annali di Matematica 203, 173–204 (2024). https://doi.org/10.1007/s10231-023-01357-4

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