Periodic-parabolic eigenvalue problems with a large parameter and degeneration

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions in periodic-parabolic problems with large parameters and degeneracies, providing new insights and methods for such degenerate eigenvalue problems.

Contribution

It introduces a novel approach to analyze the limit of eigenvalues in degenerate periodic-parabolic problems, improving previous results and connecting to semi-linear logistic equations.

Findings

Limit eigenvalue problem on degenerate domain established

Unique principal eigenvalue and eigenfunction identified in the limit

New method based on initial boundary value problem proposed

Abstract

We consider a periodic-parabolic eigenvalue problem with a non-negative potential $\lambda m$ vanishing on a non-cylindrical domain $D_m$ satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as $\lambda\to\infty$ leads a periodic-parabolic problem on $D_m$ having a unique periodic-parabolic principal eigenvalue and eigenfunction. We substantially improve a result from [Du & Peng, Trans. Amer. Math. Soc. 364 (2012), p. 6039-6070]. At the same time we offer a different approach based on a periodic-parabolic initial boundary value problem. The results are motivated by an analysis of the asymptotic behavior of positive solutions to semi-linear logistic periodic-parabolic problems with temporal and spacial degeneracies.