Lecture notes: Biological propagation via reaction-diffusion equations with nonlocal diffusion and free boundary

Abstract: These notes are based on the lectures given in a mini-course at VIASM (Vietnam Institute for Advanced Study in Mathematics) 2025 Summer School. They give a brief account of the theory (with detailed proofs) for propagation governed by a nonlocal reaction-diffusion model with free boundaries in one space dimension. The main part is concerned with a KPP reaction term, though the basic results on the existence and uniqueness of solutions as well as on the comparison principles are for more general situations. The contents are mostly taken from published recent works of the author with several collaborators, where the kernel function was assumed to be symmetric: J(x)=J(-x). When J(x) is not symmetric, significant differences may arise in the dynamics of the model, as shown in several preprints quoted in the references at the end of these notes, but many of the existing techniques can be easily extended to cover the "weakly non-symmetric case", and this is done here with all the necessary details.