THE PERIODIC LOGISTIC EQUATION WITH SPATIAL AND TEMPORAL DEGENERACIES YIHONG DU AND RUI PENG Abstract. In this article, we study the degenerate periodic logistic equation with homogeneous Neumann boundary conditions:        ∂ t u − Δu = au − b(x, t)u p in Ω × (0, ∞), ∂ ν u =0 on ∂Ω × (0, ∞), u(x, 0) = u0(x) ≥, ≡ 0 in Ω, where Ω ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary ∂Ω, a and p> 1 are constants. The function b ∈ C θ,θ/2 ( Ω × R) (0 <θ< 1) is T-periodic in t, nonnegative, and vanishes (i.e., has a degeneracy) in some subdomain of Ω × R. We examine the effects of various natural spatial and temporal degeneracies of b(x, t) on the long-time dynamical behavior of the positive solutions. Our analysis leads to a new eigenvalue problem for periodic-parabolic operators over a varying cylinder and certain parabolic boundary blow-up problems not known before. The investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the behavior of the equation, though such changes differ significantly according to whether or not there is temporal degeneracy. 1. Introduction One of the fundamental reaction-diffusion equations is the diffusive logistic equation, which is a basic model in population biology. In its simplest form, it can be written as        ∂ t u − dΔu = au − bu 2 in Ω × (0, ∞), ∂ ν u =0 on ∂ Ω × (0, ∞), u(x, 0) = u 0 (x) in Ω. This equation describes the population density u(x, t) of a species with initial density u 0 (x) and intrinsic growth rate a in a habitat Ω that has carrying capacity 1/b. The Neumann boundary condition means that the species is enclosed in Ω with no population flux across its boundary Date : March 18, 2011. 1991 Mathematics Subject Classification. 35K20, 35K60, 35J65. Key words and phrases. degenerate logistic equation, positive periodic solution, asymptotic behavior, boundary blow-up, principal eigenvalue. Y. Du (corresponding author): Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW 2351, Australia. Email: ydu@turing.une.edu.au. R. Peng: Institute of Nonlinear Complex Systems, College of Science, China Three Gorges University, Yichang City, 443002, Hubei Province, P. R. of China, and Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW 2351, Australia. Email: pengrui ¯ seu@163.com. Y. Du was partially supported by the Australian Research Council, and R. Peng was partially supported by the National Natural Science Foundation of China 10801090, 10871185, 10771032. 1