Existence of a universal attractor for a fully hyperbolic phase-field system
The article reports on the progress in investigation of the phase-field model of phase transitions with memory effects on the temperature and phase position fields. The model represents a generalization of the system studied e.g. by \textit{G. Caginalp} [Arch. Ration. Mech. Anal. 92, 205--245 (1986; Zbl 0608.35080)]. The model has been introduced in \textit{H. G. Rotstein, S. Brandon, A. Novick-Cohen} and \textit{A. Nepomnyashchy} [SIAM J. Appl. Math. 62, No. 1, 264--282 (2001; Zbl 0990.80007)] and studied by \textit{M. Grasselli} [in Kenmochi, N. (ed.), Proceedings of the international conference on free boundary problems: theory and applications, Chiba, Japan, 1999. I. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 13, 89--102 (2000; Zbl 0955.35072)], \textit{M. Grasselli} and \textit{H. G. Rotstein} [J. Math. Anal. Appl. 261, No. 1, 205--230 (2001; Zbl 0988.35117)] \textit{A. Novick-Cohen} [J. Integral Equations Appl. 14, No. 1, 73--107 (2002; Zbl 1043.45009)]. The main result of this article consists in the existence of a universal attractor for the dynamical system generated by the model equations provided the coupling function (in the text denoted as \(\lambda\)) is linear (which in fact restricts the generality of the model). The technique leading to the announced result is based on typical arguments such as dissipativity of the system (in this case rather weak), a priori estimates by the Faedo-Galerkin method and the Gronwall lemma leading to the existence of a uniform absorbing set.
scientific article; zbMATH DE number 2111541