A Sufficient Condition for the Existence of a Principal Eigenvalue for Nonlocal Diffusion Equations with Applications

TL;DR

This paper establishes a sufficient condition for the existence of a principal eigenvalue in nonlocal diffusion equations on unbounded domains, extending classical results and utilizing positive operator theory.

Contribution

It introduces a generalized Krein-Rutman theorem application for unbounded domains, providing a new criterion for principal eigenvalue existence in nonlocal diffusion equations.

Findings

A sufficient condition for principal eigenvalue existence is identified.

Generalized Krein-Rutman theorem is applicable to unbounded domains.

Positive operator theory is effective in analyzing nonlocal diffusion equations.

Abstract

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein-Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein-Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.