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 to 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 behaviour of positive solutions to semilinear logistic periodic-parabolic problems with temporal and spacial degeneracies.