Asymptotic behaviors of radially symmetric solutions to diffusion problems with Robin boundary condition in exterior domain

In this paper we study nonlinear diffusion problems of the form ut=Δu+f(u) with Robin boundary condition in exterior domain and heterogeneous environment where f(u) is a bistable term. First we prove that the radially symmetric solution converges to its equilibrium locally uniformly in the exterior domain. Then we discuss the existence of some certain equilibrium and obtain a spreading-transition-vanishing trichotomy result. Finally the behavior changes with respect to the initial data are presented.