Analysis on Fisher-KPP type nonlocal reaction diffusion equations

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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.