Stabilization of the Kawahara equation with localized damping
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, p. 102-116

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.

DOI : https://doi.org/10.1051/cocv/2009041
Classification:  35Q35,  35B40,  35Q53
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     pages = {102-116},
     doi = {10.1051/cocv/2009041},
     zbl = {1210.35215},
     mrnumber = {2775188},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 1, pp. 102-116. doi : 10.1051/cocv/2009041. http://www.numdam.org/item/COCV_2011__17_1_102_0/

[1] 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. | MR 427868 | Zbl 0229.35013

[2] N.G. Berloff and L.N. Howard, Solitary and periodic solutions for nonlinear nonintegrable equations. Stud. Appl. Math. 99 (1997) 1-24. | MR 1456147 | Zbl 0880.35105

[3] H.A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations. J. Math. Anal. Appl. 211 (1997) 131-152. | MR 1460163 | Zbl 0876.35100

[4] J.L. Bona and H. Chen, Comparison of model equations for small-amplitude long waves. Nonlinear Anal. 38 (1999) 625-647. | MR 1709992 | Zbl 0948.35108

[5] T.J. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations. SIAM J. Math. Anal. 33 (2002) 1356-1378. | MR 1920635 | Zbl 1011.35117

[6] J.M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lenghts. J. Eur. Math. Soc. 6 (2004) 367-398. | MR 2060480 | Zbl 1061.93054

[7] G.G. Doronin and N.A. Larkin, Kawahara equation in a bounded domain. Discrete Continuous Dyn. Syst., Ser. B 10 (2008) 783-799. | MR 2434910 | Zbl 1157.35437

[8] H. Hasimoto, Water waves. Kagaku 40 (1970) 401-408 [in Japanese].

[9] T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Japan 26 (1969) 1305-1318.

[10] T. Kawahara, Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33 (1972) 260-264.

[11] F. Linares and J.H. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation. ESAIM: COCV 11 (2005) 204-218. | Numdam | MR 2141886 | Zbl 1125.93006

[12] F. Linares and A.F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries with localized damping. Proc. Amer. Math. Soc. 135 (2007) 1515-1522. | MR 2276662 | Zbl 1107.93030

[13] J.L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1: Contrôlabilité Exacte, in RMA 8, Masson, Paris, France (1988). | MR 963060 | Zbl 0653.93002

[14] C.P. Massarolo, G.P. Menzala and A.F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping. Math. Meth. Appl. Sci. 30 (2007) 1419-1435. | MR 2337386 | Zbl 1114.93080

[15] 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. | MR 1878262 | Zbl 1039.35107

[16] A.F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: COCV 11 (2005) 473-486. | Numdam | MR 2148854 | Zbl 1148.35348

[17] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, USA (1983). | MR 710486 | Zbl 0516.47023

[18] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79-86. | MR 361461 | Zbl 0281.35012

[19] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33-55. | Numdam | MR 1440078 | Zbl 0873.93008

[20] L. Rosier and B.Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain. SIAM J. Contr. Opt. 45 (2006) 927-956. | MR 2247720 | Zbl 1116.35108

[21] D.L. Russell and B.Y. Zhang, Exact controllability and stabilization of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 1515-1522. | MR 1360229 | Zbl 0862.93035

[22] J.C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Diff. Equation 66 (1987) 118-139. | MR 871574 | Zbl 0631.35044

[23] G. Schneider and C.E. Wayne, The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Ration. Mech. Anal. 162 (2002) 247-285. | MR 1900740 | Zbl 1055.76006

[24] J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan 44 (1978) 663-666. | MR 489338

[25] C.F. Vasconcellos and P.N. Da Silva, Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. 58 (2008) 229-252. | MR 2436988 | Zbl 1170.35342

[26] C.F. Vasconcellos and P.N. Da Silva, Erratum of the Stabilization of the linear Kawahara equation with localized damping. Asymptotic Anal. (to appear). | Zbl 1191.35240

[27] E. Zuazua, Contrôlabilité Exacte de Quelques Modèles de Plaques en un Temps Arbitrairement Petit. Appendix in [13], 165-191.

[28] E. Zuazua, Exponential decay for the semilinear wave equation with locally distribued damping. Comm. Partial Diff. Eq. 15 (1990) 205-235. | MR 1032629 | Zbl 0716.35010