Periodic-parabolic eigenvalue problems with a large parameter and degeneration

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.

Cited by in corpus (4)

• Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials
• The weighted periodic-parabolic degenerate logistic equation
• Logistic growth in seasonally changing environments
• Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator I: Large advection