几类生物和物理模型中行波解的稳定性研究

项目名称： 几类生物和物理模型中行波解的稳定性研究

项目编号： No.11501016

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王丽娜

作者单位： 北京工商大学

项目金额： 18万元

中文摘要： 行波解的存在性稳定性问题，是非线性偏微分方程领域的经典问题. 近年来人们对它广为研究, 并取得了丰硕的成果, 但仍有很多有意义的问题未获解答. 本项目主要研究带交错扩散的拟线性退化方程组和非局部扩散方程以及带松弛的抛物双曲耦合方程组行波解的稳定性，包括谱稳定性、线性稳定性和非线性渐近稳定性. 这几类方程(组)产生于生物和物理模型中，不仅具有很强的应用背景、对应奇特的自然现象，而且是近年来偏微分方程及应用数学研究领域的国际前沿和热门的研究课题. 该项目力图在多种类型的耦合方程组的行波解稳定性及细致谱分析方面改进现有研究方法和研究理论，取得一系列创新性的研究成果，同时揭示和解释一些重要自然现象.

中文关键词： 交错扩散系统；抛物双曲耦合系统；行波解；稳定性；谱分析

英文摘要： The problem of the existence and stability of traveling waves is a classical topic in nonlinear PDE. It has been widely studied during the recent years and many results have been obtained, but there are still many interesting problems unsolved.This research project will mainly investigate the stability of the traveling waves for several classes of partial differential systems, which includes the degenerate systems with quasi-linear cross diffusion, the nonlocal diffusion equations and some coupled parabolic hyperbolic systems with relaxation. The stability here consists of spectral stability, linear stability and nonlinear asymptotic stability. The problems investigated in this project come from biological and physical model and correspond some special natural phenomena, and are also the recently pioneering problems in the research field of PDE and applied mathematics. In this project we aim to improve the related theories and research methods on the stability of traveling waves as well as the detailed spectral analysis, and try to obtain a series of important innovation research results , where some results will also reveal or explain some important natural phenomena.

英文关键词： cross diffusion systems;coupled parabolic hyperbolic systems;traveling wave;stability;spectral analysis