Periodic-parabolic eigenvalue problems with a large parameter and degeneration

We consider a periodic-parabolic eigenvalue problem with a non-negative potential λm   vanishing on a non-cylindrical domain DmDm satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as λ→∞λ→∞ leads to a periodic-parabolic problem on DmDm having a periodic-parabolic principal eigenvalue and eigenfunction which are unique in some sense. We substantially improve a result from [Du and 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.