Asymptotic behavior of degenerate logistic equations

Abstract

We analyze the asymptotic behavior of positive solutions of parabolic equations with a class of degenerate logistic nonlinearities of the type lambda u - n(x)u(rho). An important characteristic of this work is that the region where the logistic term n(.) vanishes, that is K-0 ={x : n(x) = 0}, may be non-smooth. We analyze conditions on lambda, rho, n(.) and K-0 guaranteeing that the solution starting at a positive initial condition remains bounded or blows up as time goes to infinity. The asymptotic behavior may not be the same in different parts of K-0.