Stabilization of solutions of the diffusion equation with a non-lipschitz reaction term

In this paper we are concerned with the reaction-diffusion equation u_t = Δu + f(u) in a ball of R^N with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f(u) = |u|^q-1u + |u|^p-1u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Δφ + f(φ) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.