Existence and characterization of attractors for a nonlocal reaction-diffusion equation having an energy functional

TL;DR

This paper investigates a nonlocal reaction-diffusion equation with gradient-dependent diffusion, establishing existence, uniqueness, and the existence and characterization of global attractors under weak assumptions.

Contribution

It proves the existence and uniqueness of solutions and characterizes global attractors for a nonlocal reaction-diffusion equation with gradient-dependent diffusion.

Findings

Existence and uniqueness of regular and strong solutions.

Existence of global attractors under weak assumptions.

Characterization of attractors as unstable or stable manifolds.

Abstract

In this paper we study a nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. We prove first the existence and uniqueness of regular and strong solutions. Second, we obtain the existence of global attractors in both situations under rather weak assumptions by the defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Third, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.