Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:13:31.882Z Has data issue: false hasContentIssue false
  • English
  • Français

Stabilization of the Kawahara equationwith localized damping

Published online by Cambridge University Press:  30 October 2009

Carlos F. Vasconcellos  and
Patricia N. da Silva
Carlos F. Vasconcellos
Affiliation:
Instituto de Matemática e Estatística – UERJ, 524 R. São Francisco Xavier, Sala 6016, Bloco D – CEP 20550-013, Rio de Janeiro, Brazil. [email protected]; [email protected]
Patricia N. da Silva
Affiliation:
Instituto de Matemática e Estatística – UERJ, 524 R. São Francisco Xavier, Sala 6016, Bloco D – CEP 20550-013, Rio de Janeiro, Brazil. [email protected]; [email protected]
Get access

Abstract

We study the stabilization of global solutions of theKawahara (K) equation in a bounded interval, under the effect ofa localized damping mechanism. The Kawahara equation is a modelfor small amplitude long waves. Using multiplier techniques andcompactness arguments we prove the exponential decay of the solutions of the (K) model. The proofrequires of a unique continuation theorem and the smoothing effectof the (K) equation on the real line, which are proved in this work.

Keywords

Kawahara equationstabilizationenergy decaylocalized damping
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 102 - 116
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. A 272 (1972) 47–78.
Berloff, N.G. and Howard, L.N., Solitary and periodic solutions for nonlinear nonintegrable equations. Stud. Appl. Math. 99 (1997) 1–24. CrossRef
Biagioni, H.A. and Linares, F., On the Benney-Lin and Kawahara equations. J. Math. Anal. Appl. 211 (1997) 131–152. CrossRef
Bona, J.L. and Chen, H., Comparison of model equations for small-amplitude long waves. Nonlinear Anal. 38 (1999) 625–647. CrossRef
Bridges, T.J. and Derks, G., Linear instability of solitary wave solutions of the Kawahara equation and its generalizations. SIAM J. Math. Anal. 33 (2002) 1356–1378. CrossRef
Coron, J.M. and Crépeau, E., Exact boundary controllability of a nonlinear KdV equation with critical lenghts. J. Eur. Math. Soc. 6 (2004) 367–398. CrossRef
G.G. Doronin and N.A. Larkin, Kawahara equation in a bounded domain. Discrete Continuous Dyn. Syst., Ser. B 10 (2008) 783–799.
Hasimoto, H., Water waves. Kagaku 40 (1970) 401–408 [in Japanese].
Kakutani, T. and Ono, H., Weak non-linear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Japan 26 (1969) 1305–1318. CrossRef
Kawahara, T., Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33 (1972) 260–264. CrossRef
Linares, F. and Ortega, J.H., On the controllability and stabilization of the linearized Benjamin-Ono equation. ESAIM: COCV 11 (2005) 204–218. CrossRef
Linares, F. and Pazoto, A.F., On the exponential decay of the critical generalized Korteweg-de Vries with localized damping. Proc. Amer. Math. Soc. 135 (2007) 1515–1522. CrossRef
J.L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1: Contrôlabilité Exacte, in RMA 8, Masson, Paris, France (1988).
Massarolo, C.P., Menzala, G.P. and Pazoto, A.F., On the uniform decay for the Korteweg-de Vries equation with weak damping. Math. Meth. Appl. Sci. 30 (2007) 1419–1435. CrossRef
G.P. Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping. Quarterly Applied Math. LX (2002) 111–129.
Pazoto, A.F., Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: COCV 11 (2005) 473–486. CrossRef
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, USA (1983).
Rauch, J. and Taylor, M., Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79–86. CrossRef
Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. CrossRef
Rosier, L. and Zhang, B.Y., Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain. SIAM J. Contr. Opt. 45 (2006) 927–956. CrossRef
Russell, D.L. and Zhang, B.Y., Exact controllability and stabilization of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 1515–1522. CrossRef
Saut, J.C. and Scheurer, B., Unique continuation for some evolution equations. J. Diff. Equation 66 (1987) 118–139. CrossRef
Schneider, G. and Wayne, C.E., The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Ration. Mech. Anal. 162 (2002) 247–285. CrossRef
Topper, J. and Kawahara, T., Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan 44 (1978) 663–666. CrossRef
Vasconcellos, C.F. and da Silva, P.N., Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. 58 (2008) 229–252.
C.F. Vasconcellos and P.N. da Silva, Erratum of the Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. (to appear).
E. Zuazua, Contrôlabilité Exacte de Quelques Modèles de Plaques en un Temps Arbitrairement Petit. Appendix in [13], 165–191.
Zuazua, E., Exponential decay for the semilinear wave equation with locally distribued damping. Comm. Partial Diff. Eq. 15 (1990) 205–235.