几类非线性偏微分方程组的行波解与平衡解的定性研究

项目名称： 几类非线性偏微分方程组的行波解与平衡解的定性研究

项目编号： No.11471221

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 吴雅萍

作者单位： 首都师范大学

项目金额： 65万元

中文摘要： 该项目主要研究几大类偏微分方程组的行波解、平衡解的存在性、稳定性及解的渐近性，其中包括研究几类带拟线性交错扩散项的SKT生物竞争模型与趋化性生物模型的非平凡平衡解与具奇异结构的行波解的存在性、稳定性及整体解存在性和渐近性；几类自催化反应方程组与燃烧模型的平面波解与高维柱面波解的存在性、稳定性及广义行波解的存在性；研究带空间非均匀性或非局部项的生物模型的行波解与平衡解的存在性、稳定性及一些抛物双曲耦合方程组的大强度冲击波解的渐近稳定性；还将对相关特征值问题进行深入的数值模拟和数值分析。所研究的几大类问题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、平衡解的存在性、稳定性及细致谱分析方面改进现有研究方法和研究理论，取得一系列具很高理论创新性的研究成果,同时揭示和解释一些重要自然现象。

中文关键词： 交错扩散方程组；抛物双曲耦合方程组；谱分析；行波解；平衡解

英文摘要： This research project will mainly investigate the existence and stability of the travelling waves and steady states and asymptotic behavior of solutions for several classes of partial differential systems; which includes the existence and stability of nontrivial steady states and traveling waves with singular structure and the existence and asymptotic behavior of solutions for several types of SKT competition models and chemotactic biological models with quasi-linear cross diffusion; the existence and stability of planar waves and higher dimensional waves in cylinder and the existence of generalized traveling waves for some autocatalytic reaction systems and combustion models; the existence and stability of traveling waves and steady states for some heterogeneous biological model or with nonlocal terms as well as the stability of traveling waves with large magnitude for some coupled parabolic hyperbolic system; we shall also do numerical analysis and numerical simulations on the related eigenvalue problems. The problems investigated in this project have strong background in application 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 existence and stability of steady states and traveling waves as well as the detailed spectral analysis, and try to obtain a series of important research results with higher theoretical innovation, where some results will also reveal or explain some important natural phenomena.

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