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Keywords:
Penrose-Fife model; hyperbolic equation; continuous dependence; regularity
Penrose-Fife model; hyperbolic equation; continuous dependence; regularity
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.
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