November  2008, 7(6): 1393-1414. doi: 10.3934/cpaa.2008.7.1393

Asymptotic behavior of a hyperbolic system arising in ferroelectricity

1. 

Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy

Received  June 2007 Revised  January 2008 Published  August 2008

We consider a coupled hyperbolic system which describes the evolution of the electromagnetic field inside a ferroelectric cylindrical material in the framework of the Greenberg-MacCamy-Coffman model. In this paper we analyze the asymptotic behavior of the solutions from the viewpoint of infinite-dimensional dissipative dynamical systems. We first prove the existence of an absorbing set and of a compact global attractor in the energy phase-space. A sufficient condition for the decay of the solutions is also obtained. The main difficulty arises in connection with the study of the regularity property of the attractor. Indeed, the physically reasonable boundary conditions prevent the use of a technique based on multiplication by fractional operators and bootstrap arguments. We obtain the desired regularity through a decomposition technique introduced by Pata and Zelik for the damped semilinear wave equation. Finally we provide the existence of an exponential attractor.
Citation: Sergio Frigeri. Asymptotic behavior of a hyperbolic system arising in ferroelectricity. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1393-1414. doi: 10.3934/cpaa.2008.7.1393
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