Random attractors for a stochastic nonlocal delayed reaction-diffusion equation on a semi-infinite interval

TL;DR

This paper proves the existence of random attractors for a stochastic nonlocal delayed reaction-diffusion equation modeling species evolution on a semi-infinite domain, demonstrating long-term stability under randomness.

Contribution

It introduces a novel approach to establish the existence and properties of random attractors for SNDRDEs on semi-infinite intervals with boundary conditions.

Findings

Existence of unique solutions and random dynamical systems.

Existence of bounded random absorbing sets.

Random attractors are exponentially attracting stationary solutions.

Abstract

The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic nonlocal delayed reaction-diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition on the finite end. This equation models the spatial-temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions.