We are concerned with a phase ﬁeld system consisting of two partial diﬀerential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diﬀusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann problem has been studied for such a system in [G. Canevari and P. Colli, Solvability and asymptotic analisys of a generalization of the Caginalp phase ﬁeld system, Commun. Pure Appl. Anal. 11 (2012) 1959–1982], by proving well-posedness and regularity results, as well as convergence of the problem as the coeﬃcient of the diﬀusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behaviour of the problem as the coeﬃcient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the diﬀerence of the solutions in suitable norms.
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