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

报告人：李芳 教授 （中山大学）

报告时间：2025年12月15日（星期一） 10:00-12:00

报告地点：腾讯会议：768-858-316                

主持人：王学锋 教授 香港中文大学（深圳）

校内联系人：苏远航 副教授       联系电话：84708351-8302

报告摘要： In this talk, we consider the dynamics of semilinear nonlocal diffusion equations on bounded domains with no-flux 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 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 Xueli Bai and Xuefeng Wang.

报告人简介：李芳，中山大学数学学院教授、博士生导师。本科毕业于浙江大学，博士毕业于美国明尼苏达大学。主要研究非线性椭圆和抛物方程(组)。这些方程(组)涉及生物、化学、材料等很多科学领域。近年来，关注反应扩散方程中的非局部效应等相关问题。现主持国家自然科学基金面上项目。曾主持多项国家自然科学基金项目，及上海市和广东省的省部级项目，曾参与国家自然科学基金重点项目和数学天元基金重点专项。研究成果发表在J. Math. Pures Appl., J. Funct. Anal., Calc. Var. PDE，Indiana Univ. Math. J. 等国际数学期刊上。