Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction

Global existence and asymptotic behavior of solutions for degenerate parabolic equations including u_t = λ div (|∇u|^p-2 ∇u) + |u|^q-2u(1 - |u|^r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.