Asymptotic behavior of degenerate logistic equations

We analyze the asymptotic behavior of positive solutions of parabolic equations with a class of degenerate logistic nonlinearities of the type λu−n(x)uρ. An important characteristic of this work is that the region where the logistic term n(⋅) vanishes, that is K0={x:n(x)=0}, may be non-smooth. We analyze conditions on λ, ρ, n(⋅) and K0 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 K0.