Pullback Attractors for the Non-autonomous FitzHugh-Nagumo System on Unbounded Domains

TL;DR

This paper proves the existence and boundedness of pullback attractors for a non-autonomous FitzHugh-Nagumo system on unbounded domains, even with unbounded external inputs, highlighting stability in a complex setting.

Contribution

It establishes the existence of pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains with unbounded external terms, a novel result in this context.

Findings

Pullback attractors exist for the system on unbounded domains.

Pullback attractors are uniformly bounded despite unbounded external terms.

The system's solutions exhibit asymptotic compactness under certain conditions.

Abstract

The existence of a pullback attractor is established for the singularly perturbed FitzHugh-Nagumo system defined on the entire space $R^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.