# Periodic-parabolic eigenvalue problems with a large parameter and degeneration

Preprint arXiv:1512.00485 [math.AP], September 2015
Journal of Differential Equations 261 (2016), 273–295
Original version at doi:10.1016/j.jde.2016.03.007
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.