反应扩散方程Robin边值问题解的渐近行为研究

The purpose of this project is to analyze reaction diffusion equations equipped with Robin boundary condition and Stefan free boundary condition as an ecological model to describe the population spreading or invasion of alien species. Our previous study shows that there are significant influences of Robin boundary condition on the asymptotic behavior of solutions of this model even in the simplest case of one-dimensional space and homogeneous environment, namely, the long time behavior of solutions and the free boundaries change essentially as the Robin boundary condition varies continuously. The project aims to know about whether there are similar influences of Robin boundary condition for the general model in high dimensional spaces and heterogeneous environment. Also we will give a thorough and extensive study of this model which include the convergence of bounded solutions in different Sobolev spaces, the sufficient conditions to ensure the free boundary moves to infinity or remains bounded, the asymptotic spreading speed of the front, the explanation for the influence of Robin boundary condition, free boundary condition and spatial-temporal inhomogeneity on the asymptotic behavior of the solutions and free boundaries. The research results of this project would provide some references for forecasting, prevention, risk assessment and scientific management of alien species invasion.

本项目拟研究带有Robin边界条件和Stefan自由边界条件的反应扩散方程，用此模型来描述种群传播或外来物种入侵的生态问题。在申请人的前期研究中发现，在理想的一维时空均匀环境下Robin边界条件对解的渐近行为有着显著的影响，即随着Robin边界条件中参数的连续变化解和自由边界发生本质的改变。本项目将研究一般的高维空间和时空非均匀环境下的模型，讨论Robin边界条件对解和自由边界的渐近行为的影响，并对给定的边界条件下反应扩散方程的解做深入细致的定性分析，包括有界解的收敛性，自由边界是否扩张至无穷远的充分条件，自由边界的渐近扩散速度，解和自由边界的渐近行为关于Robin边界条件、自由边界条件和非均匀时空环境的定量和定性的依赖关系等。本项目的研究成果可对外来物种入侵问题的预测、防制、风险评估及科学管理提供参考依据。

近年来带有自由边界条件的反应扩散模型研究成为生态学中的一个研究热点。比起经典的反应扩散模型，该模型在研究一些生态问题例如外来物种入侵问题等时，解的性状更贴近实际观测。而大部分研究集中在双自由边界条件或固定边界为Dirichlet或Neumann边界条件。本项目考虑的是固定边界为Robin边界条件、自由边界为Stefan条件的反应扩散方程，包括了高维情形及非线性项为非均匀时空情形，研究其解的长时间渐近行为，例如有界解收敛于其稳态解的一般收敛性、不同类型非线性项下自由边界传播至无穷远处的充分条件、自由边界的传播速度估计等。同时我们考虑了带边界层的奇异摄动椭圆问题，对有限元方法和流线扩散方法给出了三角形网格及混合网格上的收敛性分析，针对流线扩散方法设计了新型稳定化参数，对连续内罚方法证明了新的稳定性结果及超收敛结论。