Analysis on Fisher-KPP type nonlocal reaction diffusion equations

反应扩散方程是生物生物学中最经典的模型之一。自1970年以来，从PDE的角度对这些方程进行了很好的研究，包括一般的适定性结果和其他生物驱动的解行为（如有限时间爆炸、稳定解的稳定性和不稳定性、行波解、周期解等模式）。近几十年来，在对不同生物现象的建模中，引入了越来越多的具有不同类型非局部反应的方程。本文提出了具有Fisher-KPP型非局部非线性反应项的反应扩散方程（单方程和2x2系统）的新结果。对于非定域反应的总质量方程，总质量动力学起着重要的作用。研究了一类初始边值问题解的整体存在性、有限时间爆破的可能性、平稳解的存在性及其稳定性分析。研究了涉及一般积分算子的非局部项的并行结果。此外，还将进一步讨论系统的扩散驱动不稳定性。本课题将丰富现有的非局部反应扩散方程理论。

Reaction diffusion equations are one of the most classical models in biomath. Since 1970, these equations from PDE point of view have been well studied, including the general wellposedness results and other biological motivated solution behaviors (for example, finite time blow up, stability and instability of the stationary solutions, the traveling wave solutions, periodic solutions and other patterns). In the last decades, more and more equations with different type of nonlocal reactions were introduced from the modeling of different biological phenomena. This proposal aspires new results on reaction diffusion equations (a single equation and a 2x2 system) with Fisher-KPP type nonlocal nonlinear reaction terms. For the equations with total masses in the nonlocal reactions, the dynamic of total masses plays an important role. The proposed research includes the global existence of solutions to an initial boundary value problem, the possibility of finite time blow up, the existence of stationary solutions and their stability analysis. Parallel results for the nonlocal term which involves the general integral operator are pursued. In addition, the diffusion driven instability for systems will be further discussed. This proposed project will enrich the current theories on nonlocal reaction diffusion equations.