Attractors for a damped wave equation on ?3 with linear memory

Mathematical Methods in the Applied SciencesVolume 23, Issue 7 p. 633-653 Research Article Attractors for a damped wave equation on ℝ3 with linear memory Vittorino Pata, Corresponding Author Vittorino Pata [email protected] Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, ItalyDipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, Italy===Search for more papers by this author Vittorino Pata, Corresponding Author Vittorino Pata [email protected] Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, ItalyDipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, Italy===Search for more papers by this author First published: 31 March 2000 https://doi.org/10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-CCitations: 16AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract A damped semilinear hyperbolic equation on ℝ3 with linear memory is considered in a history space setting. Viewing the past history of the displacement as a variable of the system, it is possible to express the solution in terms of a strongly continuous process of continuous operators on a suitable Hilbert space. Long-time behaviour results are then discussed. In the autonomous case, the existence of a universal attractor is achieved. Copyright © 2000 John Wiley & Sons, Ltd. REFERENCES 1 Babin AV, Vishik MI. Attractors of Evolution Equations. North-Holland: Amsterdam, 1992. 2 Borini S, Pata V. Uniform attractors for a strongly damped wave equation with linear memory, Asymptotic Anal. 1999; 20: 263– 277. 3 Chepyzhov VV, Vishik MI. Nonautonomous evolution equations and their attractors. Russian J. Math. Phys. 1993; 1: 165– 190. 4 Dafermos CM. Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal. 1970; 37: 297– 308. 5 Fabrizio M, Gentili G, Reynolds DW. On a rigid linear heat conductor with memory. Int. J. Engng. Sci. 1998; 36: 765– 782. 6 Fabrizio M, Lazzari B. On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rat. Mech. Anal. 1991; 116: 139– 152. 7 Feireisl E. Attractors for semilinear wave equations on ℝ3. Nonlinear Anal. 1994; 23: 187– 195. 8 Ghidaglia JM, Temam R. Attractors for damped nonlinear hyperbolic equations. J. Math. Pure Appl. 1987; 66: 273– 319. 9 Giorgi C, Marzocchi A, Pata V. Asymptotic behavior of a semilinear problem in heat conduction with memory. NoDEA Nonlinear Differential Equations Appl. 1998; 5: 333– 354. 10 Giorgi C, Marzocchi A, Pata V. Uniform attractors for a non-autonomous semilinear heat equation with memory. Quart. Appl. Math. (to appear). 11 Graffi D. Qualche problema di elettromagnetismo. In Trends in Application of Pure Mathematics to Mechanics, Univ. of Lecce, G Fichera (ed.). Pitman: London, 1976. 12 Gurtin ME, Pipkin AC. A general theory of heat conduction with finite wave speeds. Arch. Rat. Mech. Anal. 1968; 31: 113– 126. 13 Hale J. Asymptotic Behavior of Dissipative Systems. American Mathematical Society: Province, RI, 1988. 14 Haraux A. Systèmes dynamiques dissipatifs et applications, Coll. RMA 17. Masson: Paris, 1990. 15 Liu Z, Zheng S. On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 1996; 54: 21– 31. 16 Pata V, Prouse G, Vishik MI. Traveling waves of dissipative non-autonomous hyperbolic equations in a strip. Adv. Differential Equations 1998; 3: 249– 270. 17 Pata V, Zucchi A. Attractors for a damped hyperbolic equations with linear memory (submitted). 18 Struwe M. Semilinear wave equations. Bull. Amer. Math. Soc. 1992; 26: 53– 85. 19 Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag: New York, 1988. Citing Literature Volume23, Issue710 May 2000Pages 633-653 ReferencesRelatedInformation