Existence of Traveling Wave Solutions of Competitive Systems with Density-Dependent Diffusion

報告題目：Existence of Traveling Wave Solutions of Competitive Systems with Density-Dependent Diffusion

報 告 人：王飏 副教授  山西大學

邀請人：白振國

報告時間：2026年1月10日  10：00-11：00

地點：騰訊會議 637-423-292; 密碼：123456

報告人簡介：王飏，山西大學數(shù)學與統(tǒng)計學院副教授，碩士生導師。2016年博士畢業(yè)于北京師范大學，2019年8月到2020年8月受國家留學基金委訪問學者項目資助在加拿大麥吉爾大學作訪問學者。主要研究領域為微分方程與動力系統(tǒng)，在擴散系統(tǒng)行波解和整解的存在性與穩(wěn)定性，以及最小波速的選擇機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》等刊物發(fā)表論文20余篇。先后主持國家級項目1項，省部級項目4項。

報告摘要： 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.