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Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system. (English) Zbl 1099.35021
Summary: This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable $$\chi$$, which is characterized by the presence of an inertial term multiplied by a small positive coefficient $$\mu$$. This feature is the main consequence of supposing that the response of $$\chi$$ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature $$\vartheta$$, i.e. $$\alpha (\vartheta )\sim \vartheta -1/\vartheta$$. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as $$\mu \searrow 0$$. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.
##### MSC:
 35G25 Initial value problems for nonlinear higher-order PDEs 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 80A22 Stefan problems, phase changes, etc. 35D10 Regularity of generalized solutions of PDE (MSC2000)
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