We continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea-ice growth. In a previous paper global existence and the long-time of behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here we show that the global existence of weak solutions and the long-time behavior are also studied under Neumann boundary condition. In this paper we study in space dimension lower than or equal to $3$.

The multi-phase systems has found its applications in many fields. We shall apply this approach to investigate the multi-phase dynamics of sea-ice growth. In this paper, the weak solutions existence and uniqueness of parabolic differential equations are proved. Then large-time behavior of solutions are studied, also the existence of global attractor is proved. The key tool in this article is the energy method. Our existence proof is only in one dimension.