Asymptotic behavior of the solution to the non-isothermal phase separation

Abstract

We consider the non-isothermal phase separation models of the Penrose–Fife type, which were proposed in [O. Penrose, P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62], with homogeneous Neumann boundary conditions on the nonlinear heat flux q = ∇ α ( u ) , i.e., q ⋅ n = 0 on the boundary of a region which the material occupies. Here u represents the absolute temperature. For this model, we first show that there exists a unique solution globally in time. Moreover, the ω -limit set associated with the trajectory of the unique global solution is non-empty, connected, and compact in some suitable space; as well as being composed of solutions to the steady state problem. For the stability of stationary solutions, we show that the dynamically stable solutions to the steady state problem are characterized by linearized stable solutions to the elliptic problem with a non-local term, which is equivalent to our steady state problem.