Abstract
This paper is concerned with the nonlocal dispersal equation with synchronous and asynchronous kernel functions. We study the asymptotic limiting behavior of solution when the support set of kernel function is small. Our interesting result is that the synchronous kernel function always makes the diffusion occurs in the whole domain, whereas the asynchronous kernel function can lead to the diffusion only occurs in the low-dimensional spatial domains, depending on the asynchronism of nonlocal dispersal. Moreover, we find that the solution may exhibit a quenching phenomenon for diffusion. In this situation, it is shown that the solution converges to the solution of the ODE without dispersal or it vanishes in the whole domain.
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Acknowledgements
We are very grateful to the editor and anonymous referees for their valuable comments and suggestions, which led to an improvement of our original manuscript. The initial problem was suggested by our mentor Professor Wan-Tong Li. We wish to convey our sincere thanks for his helpful comments. This work was partially supported by NSF of China (12371170) and NSF of Gansu (21JR7RA535, 21JR7RA537).
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Sun and Tao drafted the manuscript and revised the manuscript. All authors reviewed the manuscript.
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Sun, JW., Tao, W. Synchronous and Asynchronous Solutions for Some Nonlocal Dispersal Equations. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10368-5
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DOI: https://doi.org/10.1007/s10884-024-10368-5