带奇性及交错扩散项的反应扩散方程解的定性研究

This research project will mainly study the qualitative research of solutions for two types of reaction diffusion systems with strong chemical or biological background. The main research constents include the studies on the large time behavior of solutions to the degenerate Fisher equations under more general initials, the existence and stability of steady states with transition layers to S-K-T quasi-linear cross diffusion systems with small parameters and the existence and stability of non-constant steady states to quasi-linear cross diffusion systems in non-homogeneous space. In these problems, the degeneracy of the reaction term or the appearance of cross-diffusion term such that the classical research theories and methods, such as semi-group methods, spectral method, the singular perturbation method, upper-lower solution methods, the classical interpolation inequality and so on, are no longer applicable. we will find new research ideas and research methods according to the different characteristics of different problems. We expect to obtain important research results, which explain some important natural phenomena or experimental results.

该项目主要致力于两类具有强烈化学、生物背景的反应扩散方程解的定性研究。主要研究内容包括：研究退化Fisher方程在更一般初值条件下解的大时间行为；研究S-K-T型拟线性交错扩散方程组当交错扩散系数小时带边界层平衡解的存在性、稳定性，以及在非齐次空间中非常数平衡解的存在性、稳定性。由于反应项的退化性或者交错扩散项的出现使得已有的经典的研究理论和研究方法如半群方法、谱方法、奇异摄动方法、上下解方法、经典内插不等式等不再适用。我们将根据不同问题的不同特点寻找新的方法思路。期望在得到一些重要理论研究结果的同时，解释一些重要自然现象或实验成果。

本项目主要研究了退化Fisher方程在更一般初值条件下解的大时间行为，带时滞的广义Fisher方程行波解的稳定性，公共物品博弈模型整体解的存在性。对S-K-T型拟线性交错扩散方程组，当其带有小参数时我们证明了带内边界层平衡解的存在性和稳定性，对一般的强耦合抛物方程组我们证明了其全局紧吸引子的存在性及整体解的一致有界性。改进的病媒模型行波解的稳定性，Belousov-Zhabotinskii系统行波解的非线性指数稳定性以及临界Besov空间中无粘湖方程解的整体适定性.另外，还考虑了三维空间中向量积与旋度算子的高维推广问题。