Global stability in a nonlocal reaction-diffusion equation

Abstract

We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman--Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

References in corpus (3)

• A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• The statistical dynamics of a spatial logistic model and the related kinetic equation
• Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation

Cited by in corpus (2)

• Spatial Propagation in Nonlocal Dispersal Fisher-KPP Equations
• Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media