时滞反应扩散方程中的分支与空间非均匀解

This program is devoted to the study of spatially inhomogeneous solutions driven by diffusion and delay, including steady states and time-periodic solutions, by investigating the bifurcations of delayed reaction diffusion equations. It consists of, for no flux boundary problem, the existence, stability and global continuation of inhomogeneous solutions bifurcated from homogeneous steady states as well as the high codimensional bifurcations of homogeneous steady states and, for zero Dirichlet boundary problem, the existence of inhomogeneous steady states as well as the local and global existence of Hopf bifurcations from such steady states. For no flux boundary problem, there are mumerical simulations showing that such system can have spatially inhomogeneous time periodic or quasi-periodic solutions, which can not be seen in its kinetic system. But, the theoretical studies on such phenomena are few in literature. For zero Dirichlet boundary problem, non-zero steady states are spatially inhomogeneous and lack of closed forms, which gives rise to difficulties for the study of bifurcations around them. To conduct such program, one not only needs the existing theory and tools of bifurcations but also require new ideas and methods.

本项目拟通过研究时滞反应扩散方程的分支问题来考察由空间扩散和时滞导致的空间非均匀解（包括稳态解和周期解）。研究内容包括：对齐次Neumann边界条件下的具时滞或空间结构的反应扩散方程，研究由空间均匀稳态解附近的分支产生的空间非均匀周期解的存在性、稳定性和全局延拓性以及空间均匀稳态解附近的高余维分支；对齐次 Dirichlet边界条件下的时滞反应扩散方程，研究空间非均匀稳态解的存在性稳定性以及它附近的局部及全局Hopf分支的存在性。对齐次Neumann边界条件下的时滞反应扩散方程，有数值模拟显示存在空间非均匀时间上是周期或拟周期的解，这是相应的反应方程所不具有的，然而，此方向的理论研究还比较少。对齐次Dirichlet边界条件下的方程，非零稳态解都是空间非均匀的，这种稳态解本身的空间非均匀性导致了对其分支研究的困难性。本项目不仅需要应用和发展已有的分支问题的研究工具，同时需要新的思想和方法。

基本完成了项目研究计划。主要结果有：1.在Neumann边界条件下得到了两个模型由Hopf分支和Turing-Hopf分支产生的空间非均匀解的存在性和稳定性，数值模拟结果显示这种空间非均匀周期解会长时间存在。2.应用反应扩散方程分支理论分析了两类捕食者-食饵模型的分支问题；3.得到了两个区域上的果蝇模型在Dirichlet边界条件下的局部和全局Hopf分支存在性；4.对纯量方程证明了时滞不会使不稳定的稳态解变稳定。共发表论文4篇，出版专著1部。