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

报告人：白学利， 西北工业大学

时间：9月17日19点

腾讯会议：853-491-582

摘要：In this talk, we will investigate the dynamics of semilinear nonlocal diffusion equations on bounded domains with no flux boundary conditions, specially 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 one dimensional 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 joint work with Fang Li and Xuefeng Wang.