The periodic logistic equation with spatial and temporal degeneracies

Abstract

In this article, we study the degenerate periodic logistic equation with homogeneous Neumann boundary conditions: { ∂ t u − Δ u = a u − b ( x , t ) u p a m p ; in Ω × ( 0 , ∞ ) , ∂ ν u = 0 a m p ; on ∂ Ω × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ≥ , ≢ 0 a m p ; in Ω , \begin{equation*} \begin {cases} \partial _t u-\Delta u=a u-b(x,t)u^p & \text {in $\Omega \times (0,\infty )$},\\ \partial _\nu u=0 & \text {on $\partial \Omega \times (0,\infty )$},\\ u(x,0) = u_0(x)\geq , \nequiv 0 & \text {in $\Omega $}, \end{cases} \end{equation*} where Ω ⊂ R N ( N ≥ 2 ) \Omega \subset \mathbb {R}^N\ (N\geq 2) is a bounded domain with smooth boundary ∂ Ω \partial \Omega , a a and p > 1 p>1 are constants. The function b ∈ C θ , θ / 2 ( Ω ¯ × R ) b\in C^{\theta ,\theta /2}(\overline \Omega \times \mathbb {R}) ( 0 > θ > 1 ) (0>\theta >1) is T-periodic in t t , nonnegative, and vanishes (i.e., has a degeneracy) in some subdomain of Ω × R \Omega \times \mathbb {R} . We examine the effects of various natural spatial and temporal degeneracies of b ( x , t ) 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.