Some effects of nonlocal diffusion on the solutions of Fisher-KPP equations in disconnected domains
The paper studies an elliptic Fisher-KPP equation, with diffusion driven by the fractional Laplacian. The equation is set on a fragmented domain, i.e., the -- possibly, infinite -- union of components, resulting in a possibly unbounded domain. Under spectral and geometrical assumptions on such domain, the authors prove existence and uniqueness of solutions to the equation when coupled with homogeneous external Dirichlet conditions. It has to be stressed how existence and uniqueness are peculiar consequences of the nonlocality of the leading operator, allowing for connected components to exchange information with each other. The analysis is complemented by a double-sided estimate on the solutions and the inspection of the behaviour (especially continuity) of the principal eigenvalue in the particular case of dimension one and of a domain made up of two patches.
scientific article; zbMATH DE number 7620716
scientific article