Asymptotic behavior of a hyperbolic system arising in ferroelectricity

Previous Article
Stabilisation of differential inclusions and PDEs without uniqueness by noise
CPAA Home
This Issue
Next Article
Linearization of smooth planar vector fields around singular points via commuting flows

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

Abstract
Related Papers

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.

Keywords: exponential attractor., damped semilinear wave equation, Absorbing sets, global attractor.

Mathematics Subject Classification: 35B40, 35B41, 35L05, 35Q60, 35R15, 37L05, 37L2.

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

[1]	Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
[2]	Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305
[3]	Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[4]	Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[5]	Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019179
[6]	Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559
[7]	Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
[8]	Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[9]	D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[10]	Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022
[11]	Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[12]	Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[13]	Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121
[14]	Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301
[15]	Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[16]	Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3799-3825. doi: 10.3934/dcds.2015.35.3799
[17]	Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695
[18]	Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[19]	John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[20]	Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences