交错扩散与非经典扩散方程组解的定性研究

This research project will mainly investigate the qualitative behavior of solutions for several types of quasilinear diffusion systems and non-classical diffusion systems; the main research topics include the qualitative studies on the asymptotic behavior of solutions, the existence and stability of steady states and traveling waves with singular structure and the existence of blow-up solutions for several types of quasi-linear cross diffusion systems; the existence and stability of non-monotone waves, higher dimensional cylinder waves and periodic waves and the existence of generalized traveling waves for some diffusion models; the existence and stability of generalized traveling waves and steady states for some biological models in heterogenerous environment or with nonlocal terms; and the stability of planar waves with large magnitude and periodic wave for some coupled parabolic hyperbolic systems. The problems investigated in this project have strong background in application and correspond to some special pattern formation and wave phenomena in applied models, and some problems are the recently pioneering or attracting 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 wave solutions and steady states as well as the asymptotic behavior of solutions, and try to obtain a series of important research results with higher theoretical innovation, which will also reveal or explain some important natural phenomena.

该项目主要研究几大类拟线性交错扩散方程组和非经典扩散方程组解的定性性质，主要研究课题包括：研究几类拟线性交错扩散方程组的解的渐近性、具奇异结构的平衡解和行波解的存在性稳定性及爆破解的存在性；研究几类扩散模型的非单调波、高维柱面波解和周期波解的存在性稳定性和广义行波解的存在性；研究在非均匀环境或带非局部项的生物扩散模型的广义行波解与平衡解的存在性稳定性；研究几类抛物双曲耦合方程组的大强度平面波解和周期波的渐近稳定性。所研究的几类问题不仅具有很强的应用背景、对应实际模型中奇特的斑图和波现象，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或很受关注的研究课题。该项目力图在多种类型的波解和平衡解的存在性、稳定性和解的渐近性研究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和应用性的研究成果,同时揭示或解释一些重要自然现象。

该项目取得的主要成果和研究进展如下：应用多种研究理论包括特殊奇异摄动法和高阶渐近展开、特殊的变换、非经典分叉理论、Evans 函数法、Lyapunov-Schmitz分解法与细致的谱分析、特殊上下解构造和数值模拟巧妙融合，并改进相关研究框架和研究技巧及数值模拟方法, 对一类重要的带奇性的Keller-Segel趋化模型得到了脉冲细菌波解在特殊加权空间中的谱稳定性/不稳定性; 对一类具logistic增长项的趋化性生物模型证明了行波解的谱稳定性；对几类反应扩散模型研究了单调和无穷震荡波在指数加权空间的渐近稳定性；对一类带小参数的SKT交错扩散模型研究了具内边界层的波弱相互作用和渐近移动速度；对具大交错扩散系数几类SKT型交错扩散方程组和相应的极限方程组得到了具特殊爆破结构的平衡解的存在性和不稳定性及一类大尖峰平衡解的稳定性；证明了几类交错扩散模型具局部和整体分叉结构的平衡解存在性和稳定性；对一类高维柱形域上带非局部项的Fisher方程证明了具非临界波速的柱形域波的谱稳定性和研究了一般初值解的渐近性；对几类带非局部时滞项的应用模型证明了行波解的稳定性。所研究的模型和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或热门的研究课题。上述代表性研究成果均为全新的研究结果，并在研究框架、研究方法和研究技巧上有本质创新，部分成果解决了相关领域很受关注的公开问题，得到的丰富深刻的定性和数值研究成果同时揭示和解释了一些重要自然现象。