# Global stability in a nonlocal reaction-diffusion equation

## TL;DR

This paper investigates the stability of stationary solutions in a class of nonlocal reaction-diffusion equations, introducing new conditions for stability and applying them to ecological models with stochastic initial conditions.

## Contribution

It establishes the Feynman--Kac formula for Lévy processes with time-dependent potentials and derives novel stability criteria for solutions of nonlocal semilinear parabolic equations.

## Key findings

- Conditions for asymptotic stability of zero solution
- Stability criteria for positive stationary solutions in ecological models
- Analysis of stability with random initial conditions

## 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 L\'{e}vy 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.