This paper deals with a free boundary problem for a reaction-diffusion equation with moving boundary, whose dynamics is governed by the Stefan condition. We will mainly discuss the problem for the case of multi-stable nonlinearity, which is a function with a multiple number of positive stable equilibria. The first result is concerned with the classi cation of solutions in accordance with large-time behaviors. As a consequence, one can observe a multiple number of spreading phenomena corresponding for each positive stable equilibrium. Here it is seen that there exists a certain group of spreading solutions whose element accompanies a propagating terrace. We will derive sharp asymptotic estimates of free boundary and profile of every spreading solution including spreading one with propagating terrace.
