讲座题目：Existence of Traveling Wave Solutions of Lotka-Volterra Competitive Systems with Density-Dependent Diffusion

时间：2025年7月15日下午15:00

地点：理学楼A301

报告人：王飏 山西大学 副教授

报告人简介：王飏，山西大学数学与统计学院副教授，硕士生导师。2016年博士毕业于北京师范大学，2019年8月到2020年8月受国家留学基金委访问学者项目资助在加拿大麦吉尔大学作访问学者。主要研究领域为微分方程与动力系统，在扩散系统行波解和整解的存在性与稳定性，以及最小波速的选择机制研究中得一系列成果。已在《Nonlinearity》《Proc. Roy. Soc. Edinburgh Sect. A》《J. Math. Phys》《ZAMM Z. Angew. Math. Mech》《Z. Angew. Math. Phys》等刊物发表论文20余篇。先后主持国家级项目1项，省部级项目4项。

讲座内容：

In this talk, we focus on the existence of traveling wave solutions for two-species competitive systems with density-dependent diffusion. Since density-dependent diffusion is a form of nonlinear diffusion that degenerates at the origin, traditional methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are inapplicable. To address this diffusion degeneracy, we construct a nonlinear invariant region near the origin. Utilizing the method of phase plane analysis, we demonstrate the existence of traveling wave solutions that connect the origin to the unique coexistence state when the wave speed c exceeds a certain positive threshold value. Furthermore, when one species exhibits density-dependent diffusion and the other exhibits linear diffusion, we employ the phase transform and the central manifold theorem to establish the existence of a minimal wave speed c*, which is less than the threshold value. For wave speeds c≥c*, traveling wave solutions that connect the origin to the unique coexistence state exist. Notably, at c=c*, we observe that one component of the traveling wave solution is of a sharp type, while the other is smooth. This phenomenon is distinct from what is observed in systems with linear diffusion or scalar equations. This is a joint work with Xuanyu Lv, Fan Liu and Xiaoguang Zhang.

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理学院

2025年6月25日