Stability for steady-state patterns in phase field dynamics associated with total variation energies

In this paper, we shall deal with a mathematical model to represent the dynamics of solid-liquid phase transitions, which take place in a two-dimensional bounded domain. This mathematical model is formulated as a coupled system of two kinetic equations.
    The first equation is a kind of heat equation, however a time-relaxation term is additionally inserted in the heat flux. Since the additional term guarantees some smoothness of the velocity of the heat diffusion, it is expected that the behavior of temperature is estimated in stronger topology than that as in the usual heat equation.
    The second equation is a type of the so-called Allen-Cahn equation, namely it is a kinetic equation of phase field dynamics derived as a gradient flow of an appropriate functional. Such functional is often called as "free energy'', and in case of our model, the free energy is formulated with use of the total variation functional. Therefore, the second equation involves a singular diffusion, which formally corresponds to a function of (mean) curvature on the free boundary between solid-liquid states (interface). It implies that this equation can be a modified expression of Gibbs-Thomson law.
    In this paper, we will focus on the geometry of the pattern drawn by solid-liquid phases in steady-state (steady-state pattern), which will be expected to have some stability in dynamical system generated by our mathematical model. Consequently, various geometric patterns, parted by gradual curves, will be shown as representative examples of such steady-state patterns.