非局部反应扩散方程行波解的存在性和定性分析

It has realistic meaning to study the existence, the uniqueness and the stability of the traveling waves for the nonlocal reaction diffusion equations with time delay. Firstly, the existence and the comparison principle of solutions for the reaction diffusion equation with nonlocal diffusion term and time delay are established by considering the theory of functional differential equation. It is helpful to study the existence of traveling waves by constructing a pair of upper-lower solutions and Schauder's fixed point theorem. Furthermore, we develop the existence of traveling waves for the predator-prey model with nonlocal diffusion term and time delay. Finally, we obtain the uniqueness and stability of traveling waves for the epidemic model and the predator-prey model by using the squeezing method and spectral analysis of linearized operator.

研究带有非局部扩散项和时滞的反应扩散方程行波解的存在性、唯一性和稳定性是一项具有现实意义的工作。本项目计划通过抽象泛函微分方程的理论给出带有非局部扩散项和时滞的反应扩散方程解的存在性和比较原理，再通过构造合适的上下解和Schauder不动点定理来得到其行波解的存在性，进而考虑带有非局部扩散项和时滞的捕食模型行波解的存在性。最后，对已知行波解存在性的传染病模型和捕食模型，利用挤压原理或者谱理论分析得到该行波解的唯一性和稳定性。

本项目主要研究带有非局部扩散项和时滞项的反应扩散方程行波解的存在性、唯一性和稳定性，该课题是一项具有现实意义的工作。本项目通过抽象泛函微分方程的理论给出带有非局部扩散项和时滞的反应扩散方程解的存在性和比较原理，再通过构造合适的上下解和Schauder不动点定理来得到其行波解的存在性，进而考虑带有非局部扩散项和时滞的捕食模型行波解的存在性。最后，对已知行波解存在性的传染病模型和捕食模型，利用挤压原理或者谱理论分析得到该行波解的唯一性和稳定性。本项目对于控制传染病的传播和生态平衡科研提供一定的理论依据，具有一定的实际意义。