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Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation


Authors: Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti and Flávio A. F. Nascimento
Journal: Proc. Amer. Math. Soc. 141 (2013), 3183-3193
MSC (2010): Primary 35L05, 34Dxx, 35A27
DOI: https://doi.org/10.1090/S0002-9939-2013-11869-1
Published electronically: May 29, 2013
Erratum: Proc. Amer. Math. Soc. 145 (2017), 4097-4097.
MathSciNet review: 3068971
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Abstract: We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold $ (M, \bf g)$ subject to locally distributed viscoelastic effects on a subset $ \omega \subset M$. Assuming that the well-known geometric control condition $ (\omega , T_0)$ holds and supposing that the relaxation function is bounded by a function that decays exponentially to zero, we show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero. We give a new geometric proof extending the prior results in the literature from the Euclidean setting to compact Riemannian manifolds (with or without boundary).


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  • [BAR-LE-RAU] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992), no. 5, 1024-1065. MR 1178650 (94b:93067)
  • [BEL] M. Bellassoued, Decay of solutions of the elastic wave equation with a localized dissipation, Annales de la Faculté des Sciences de Toulouse, XII (3), 2003, 267-301. MR 2030088 (2005b:35169)
  • [CA-OQ] M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42 (2003), no. 4, 1310-1324. MR 2044797 (2005d:35257)
  • [CA-DO-FU-SO] M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping, Methods Appl. Anal. 15(4) (2008), 405-426. MR 2550070 (2010m:35285)
  • [CA-DO-FU-SO-1] M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping, Transactions of AMS 361(9) (2009), 4561-4580. MR 2506419 (2010e:58030)
  • [CA-DO-FU-SO-2] M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result. Arch. Ration. Mech. Anal. 197 (2010), no. 3, 925-964. MR 2679361 (2012a:35331)
  • [CHR] H. Christianson, Semiclassical non-concentration near hyperbolic orbits,
    J. Funct. Anal. 246 (2007), no. 2, 145-195. MR 2321040 (2008k:58058)
  • [DA] C. M. Dafermos, Asymptotic behavior of solutions of evolution equations. Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), pp. 103-123, Publ. Math. Res. Center Univ. Wisconsin, 40, Academic Press, New York-London, 1978. MR 513814 (80i:35019)
  • [DA-LA-TO] M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partialy supported nonlinear boundary dissipation without growth restrictions, DCDS-S, 2, 1 (2009). MR 2481581 (2010f:35227)
  • [DEH-LE-ZUA] B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Anna. Sci. Ec. Norm. Super. 36, 525-551 (2003). MR 2013925 (2004i:35223)
  • [HI] M. Hitrik, Expansions and eigenfrequencies for damped wave equations, Journées équations aux Dérivées Partielles (Plestin-les-Grèves, 2001), Exp. No. VI, 10 pp., Univ. Nantes, Nantes, 2001. MR 1843407 (2002h:58050)
  • [GE] P. Gerárd, Microlocal defect measures, Com. Par. Diff. Eq. 16 (1991), 1761-1794. MR 1135919 (92k:35027)
  • [GES-MES] A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Analysis: Real World Applications 13 (2012), 476-485. MR 2846857 (2012i:35213)
  • [LE] G. Lebeau, Equations des ondes amorties, Algebraic Geometric Methods in Maths. Physics, pp. 73-109, 1996. MR 1385677 (97i:58173)
  • [MAR] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complutense 12(1) (1999), 251-283. MR 1698906 (2000d:93037)
  • [MI] L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Control Optim. 41 (2002), no. 5, 1554-1566. MR 1971962 (2004b:93115)
  • [RI-SA] J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials. Quart. Appl. Math. 59 (2001), no. 3, 557-578. MR 1848535 (2002f:35038)
  • [NA] M. Nakao, Decay and global existence for nonlinear wave equations with localized dissipations in general exterior domains. New trends in the theory of hyperbolic equations, 213-299, Oper. Theory Adv. Appl., 159, Birkhäuser, Basel (2005). MR 2175918 (2006i:35250)
  • [NA1] M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr. 278(7-8) (2005), 771-783. MR 2141956 (2006a:35185)
  • [RAU-TAY] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds. Comm. Pure Appl. Math. 28(4) (1975), 501-523. MR 0397184 (53:1044a)
  • [QIN] T. Qin, Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications. J. Math. Anal. Appl. 233 (1999), no. 1, 130-147. MR 1684377 (2000a:45023)
  • [TOUN] D. Toundykov, Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary, Nonlinear Analysis T. M. A. 67(2) (2007), 512-544. MR 2317185 (2008f:35257)
  • [TRI-YAO] R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, Appl. Math. and Optim. 46 (Sept./Dec. 2002), 331-375. Special issue dedicated to J. L. Lions. MR 1944764 (2003j:93042)
  • [ZUA] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations 15 (1990), no. 2, 205-235. MR 1032629 (91b:35076)

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Additional Information

Marcelo M. Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil
Email: mmcavalcanti@uem.br

Valéria N. Domingos Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil
Email: vndcavalcanti@uem.br

Flávio A. F. Nascimento
Affiliation: Department of Mathematics, State University of Ceará-FAFIDAM, 62930-000, Limoeiro do Norte, CE, Brazil
Email: flavio.falcao@uece.br

DOI: https://doi.org/10.1090/S0002-9939-2013-11869-1
Keywords: Wave equation, compact Riemannian manifold, viscoelastic distributed damping
Received by editor(s): November 28, 2011
Published electronically: May 29, 2013
Additional Notes: Research of the first author was partially supported by the CNPq Grant 300631/2003-0
Research of the second author was partially supported by the CNPq Grant 304895/2003-2
The third author, a doctorate student at the State University of Maringá, was partially supported by a grant of CNPq, Brazil
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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