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Global attractor for a parabolichyperbolic PenroseFife phase field system
Stability for steadystate patterns in phase field dynamics associated with total variation energies
1.  Department of Applied Mathematics, Faculty of Engineering, Kobe University, 11 Rokkodai, Nada, Kobe, 6578501, Japan 
The first equation is a kind of heat equation, however a timerelaxation 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 socalled AllenCahn 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 solidliquid states (interface). It implies that this equation can be a modified expression of GibbsThomson law.
In this paper, we will focus on the geometry of the pattern drawn by solidliquid phases in steadystate (steadystate 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 steadystate patterns.
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