We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.
Mots clés : Kawahara equation, stabilization, energy decay, localized damping
@article{COCV_2011__17_1_102_0,
author = {Vasconcellos, Carlos F. and da Silva, Patricia N.},
title = {Stabilization of the {Kawahara} equation with localized damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {102--116},
publisher = {EDP-Sciences},
volume = {17},
number = {1},
year = {2011},
doi = {10.1051/cocv/2009041},
mrnumber = {2775188},
zbl = {1210.35215},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009041/}
}
TY - JOUR AU - Vasconcellos, Carlos F. AU - da Silva, Patricia N. TI - Stabilization of the Kawahara equation with localized damping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 102 EP - 116 VL - 17 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009041/ DO - 10.1051/cocv/2009041 LA - en ID - COCV_2011__17_1_102_0 ER -
%0 Journal Article %A Vasconcellos, Carlos F. %A da Silva, Patricia N. %T Stabilization of the Kawahara equation with localized damping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 102-116 %V 17 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009041/ %R 10.1051/cocv/2009041 %G en %F COCV_2011__17_1_102_0
Vasconcellos, Carlos F.; da Silva, Patricia N. Stabilization of the Kawahara equation with localized damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 102-116. doi : 10.1051/cocv/2009041. http://www.numdam.org/articles/10.1051/cocv/2009041/
[1] , and , Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. A 272 (1972) 47-78. | MR | Zbl
[2] and , Solitary and periodic solutions for nonlinear nonintegrable equations. Stud. Appl. Math. 99 (1997) 1-24. | MR | Zbl
[3] and , On the Benney-Lin and Kawahara equations. J. Math. Anal. Appl. 211 (1997) 131-152. | MR | Zbl
[4] and , Comparison of model equations for small-amplitude long waves. Nonlinear Anal. 38 (1999) 625-647. | MR | Zbl
[5] and , Linear instability of solitary wave solutions of the Kawahara equation and its generalizations. SIAM J. Math. Anal. 33 (2002) 1356-1378. | MR | Zbl
[6] and , Exact boundary controllability of a nonlinear KdV equation with critical lenghts. J. Eur. Math. Soc. 6 (2004) 367-398. | MR | Zbl
[7] and , Kawahara equation in a bounded domain. Discrete Continuous Dyn. Syst., Ser. B 10 (2008) 783-799. | MR | Zbl
[8] , Water waves. Kagaku 40 (1970) 401-408 [in Japanese].
[9] and , Weak non-linear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Japan 26 (1969) 1305-1318.
[10] , Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33 (1972) 260-264.
[11] and , On the controllability and stabilization of the linearized Benjamin-Ono equation. ESAIM: COCV 11 (2005) 204-218. | Numdam | MR | Zbl
[12] and , On the exponential decay of the critical generalized Korteweg-de Vries with localized damping. Proc. Amer. Math. Soc. 135 (2007) 1515-1522. | MR | Zbl
[13] , Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1: Contrôlabilité Exacte, in RMA 8, Masson, Paris, France (1988). | MR | Zbl
[14] , and , On the uniform decay for the Korteweg-de Vries equation with weak damping. Math. Meth. Appl. Sci. 30 (2007) 1419-1435. | MR | Zbl
[15] , and , Stabilization of the Korteweg-de Vries equation with localized damping. Quarterly Applied Math. LX (2002) 111-129. | MR | Zbl
[16] , Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: COCV 11 (2005) 473-486. | Numdam | MR | Zbl
[17] , Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, USA (1983). | MR | Zbl
[18] and , Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79-86. | MR | Zbl
[19] , Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33-55. | Numdam | MR | Zbl
[20] and , Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain. SIAM J. Contr. Opt. 45 (2006) 927-956. | MR | Zbl
[21] and , Exact controllability and stabilization of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 1515-1522. | MR | Zbl
[22] and , Unique continuation for some evolution equations. J. Diff. Equation 66 (1987) 118-139. | MR | Zbl
[23] and , The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Ration. Mech. Anal. 162 (2002) 247-285. | MR | Zbl
[24] and , Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan 44 (1978) 663-666. | MR
[25] and , Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. 58 (2008) 229-252. | MR | Zbl
[26] and , Erratum of the Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. (to appear). | Zbl
[27] , Contrôlabilité Exacte de Quelques Modèles de Plaques en un Temps Arbitrairement Petit. Appendix in [13], 165-191.
[28] , Exponential decay for the semilinear wave equation with locally distribued damping. Comm. Partial Diff. Eq. 15 (1990) 205-235. | MR | Zbl
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