Existence of a universal attractor for a fully hyperbolic phase-field system
type: Item
instance of: scholarly article
title: Existence of a universal attractor for a fully hyperbolic phase-field system
zbMATH Open document ID: 1063.35038
author: Maurizio Grasselli
author: Vittorino Pata
published in: Journal of Evolution Equations
publication date: 2004-10-28T00:00:00Z
review text: 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.
reviewed by: Michal Beneš
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 80A22
zbMATH DE Number: 2111541
zbMATH Keywords: memory effects
zbMATH Keywords: absorbing sets
zbMATH Keywords: dissipativity
zbMATH Keywords: Faedo-Galerkin method
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2111541