A nonlocal diffusion model with free boundaries in a spatially heterogeneous environment

In this paper, we study a nonlocal diffusion problem with free boundaries in a spatially heterogeneous environment. Since both the diffusion and reaction terms include nonlocal operators that do not satisfy the local Lipschitz continuity condition, many traditional methods are not directly applicable. We introduce new techniques to address these challenges. We first establish the global existence and uniqueness of the solution. By studying the principal eigenvalue of the problem, we obtain the spreading-vanishing dichotomy and the corresponding criteria. Under a suitable condition, we determine the long-term behavior of the solution when spreading occurs.