勷勤数学•专家报告

题      目：Existence and nonexistence of stable patterns in semilinear nonlocal diffusion equations

报  告  人： 白学利 教授  (邀请人：李颖花)

                                    西北工业大学

时      间： 1月23日  15：30-16:30

地     点：数科院东楼401

报告人简介:

        白学利，西北工业大学理学院教授，2012年在大连理工大学获得博士学位。2012年-2015年在华东师范大学偏微分方程中心跟随倪维明教授进行博士后研究。2017年-2018年受洪堡基金资助在德国帕德博恩大学同数学家Michael Winkler进行合作研究。主要研究方向为生物数学与偏微分方程。 在JEMS,JFA, IUMJ, CVPDE, JDE等数学期刊发表论文多篇，主持国家自然科学基金等多项科研项目。

摘      要：

        This paper investigates the dynamics of semilinear nonlocal diffusion equations on bounded domains with noflux boundary conditions, specifically focusing on the existence and stability of non-constant steady states, referred to as patterns. According to the results of Casten, Holland, and Matano regarding semilinear local diffusion equations, we know that stable patterns do not exist in convex domains, while they do emerge in dumbbell-shaped geometries, particularly when the kinetic term is bistable. We extend these findings to nonlocal diffusion analogs, demonstrating the absence of stable smooth patterns in both onedimensional intervals and multi-dimensional balls. In addition, we construct discontinuous, asymptotically stable patterns when the kinetic term is bistable. Our results reveal a significant principle: large nonlocal diffusion tends to destabilize patterns, whereas weak nonlocal diffusion stabilizes them, especially in cases with bistable kinetic terms. Importantly, the geometry of the domain appears to play a less critical role in this process of stabilization. This is a joint work with Fang Li(Sun Yat-sen University) and Xuefeng Wang(The Chinese University of Hong Kong，Shenzhen).

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