  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.
