Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Uniform stabilization of a nonlinear dispersive system


Authors: A. F. Pazoto and G. R. Souza
Journal: Quart. Appl. Math. 72 (2014), 193-208
MSC (2010): Primary 93D15, 93B05; Secondary 35B40, 35Q53
DOI: https://doi.org/10.1090/S0033-569X-2013-01343-1
Published electronically: December 6, 2013
MathSciNet review: 3185138
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. A locally exponential decay result is derived.


References [Enhancements On Off] (What's this?)

  • [1] Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett. 31 (1973), 125-127. MR 0406176 (53 #9968)
  • [2] Eduardo Alarcon, Jaime Angulo, and Jose F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal. 36 (1999), no. 8, Ser. A: Theory Methods, 1015-1035. MR 1684527 (2000b:35212), https://doi.org/10.1016/S0362-546X(97)00724-4
  • [3] E. Bisognin, V. Bisognin, and G. Perla Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries equations with localized damping, Adv. Differential Equations 8 (2003), no. 4, 443-469. MR 1972596 (2004c:35349)
  • [4] Jerry Bona and Ragnar Winther, The Korteweg-de Vries equation, posed in a quarter-plane, SIAM J. Math. Anal. 14 (1983), no. 6, 1056-1106. MR 718811 (85c:35076), https://doi.org/10.1137/0514085
  • [5] Jerry L. Bona, Gustavo Ponce, Jean-Claude Saut, and Michael M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys. 143 (1992), no. 2, 287-313. MR 1145797 (93e:35086)
  • [6] Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Andrei Faminskii, and Fábio Natali, Decay of solutions to damped Korteweg-de Vries type equation, Appl. Math. Optim. 65 (2012), no. 2, 221-251. MR 2891222, https://doi.org/10.1007/s00245-011-9156-7
  • [7] Eduardo Cerpa and Ademir F. Pazoto, A note on the paper ``On the controllability of a coupled system of two Korteweg-de Vries equations'' [MR2561938], Commun. Contemp. Math. 13 (2011), no. 1, 183-189. MR 2772582 (2012a:93006), https://doi.org/10.1142/S021919971100418X
  • [8] M. Davila, On the unique continuation property for a coupled system of Korteweg-de Vries equations, PhD Thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Brazil, (1994).
  • [9] J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math. 70 (1984), no. 3, 235-258. MR 742590 (85i:76013)
  • [10] Camille Laurent, Lionel Rosier, and Bing-Yu Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations 35 (2010), no. 4, 707-744. MR 2753618 (2012d:93024), https://doi.org/10.1080/03605300903585336
  • [11] F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal. 3 (2004), no. 3, 417-431. MR 2098292 (2005h:35308), https://doi.org/10.3934/cpaa.2004.3.417
  • [12] F. Linares and A. F. Pazoto, Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane, J. Differential Equations 246 (2009), no. 4, 1342-1353. MR 2488687 (2010i:35337), https://doi.org/10.1016/j.jde.2008.11.002
  • [13] F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc. 135 (2007), no. 5, 1515-1522 (electronic). MR 2276662 (2008e:35168), https://doi.org/10.1090/S0002-9939-07-08810-7
  • [14] C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations 22 (2009), no. 1-2, 53-68. MR 2483012 (2010h:35349)
  • [15] C. P. Massarolo, G. P. Menzala, and A. F. Pazoto, Uniform stabilization of a class of coupled systems of KdV equations with localized damping, Quart. Appl. Math. 69 (2011), no. 4, 723-746. MR 2893997 (2012m:35292), https://doi.org/10.1090/S0033-569X-2011-01245-6
  • [16] C. P. Massarolo, G. P. Menzala, and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci. 30 (2007), no. 12, 1419-1435. MR 2337386 (2008c:35282), https://doi.org/10.1002/mma.847
  • [17] G. Perla Menzala, C. F. Vasconcellos, and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. 60 (2002), no. 1, 111-129. MR 1878262 (2002j:35273)
  • [18] Sorin Micu and Jaime H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000, pp. 1020-1024. MR 1786022
  • [19] Sorin Micu, Jaime H. Ortega, and Ademir F. Pazoto, On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math. 11 (2009), no. 5, 799-827. MR 2561938 (2010m:93017), https://doi.org/10.1142/S0219199709003600
  • [20] L. Molinet and D. Pilod, The Cauchy problem for the Benjamin-On equation in $ L^2$ revisited, Analysis and PDE, to appear.
  • [21] Dugan Nina, Ademir F. Pazoto, and Lionel Rosier, Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain, Math. Control Relat. Fields 1 (2011), no. 3, 353-389. MR 2846090 (2012k:93154), https://doi.org/10.3934/mcrf.2011.1.353
  • [22] Ademir Fernando Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var. 11 (2005), no. 3, 473-486 (electronic). MR 2148854 (2006b:35292), https://doi.org/10.1051/cocv:2005015
  • [23] Lionel Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33-55 (electronic). MR 1440078 (98d:93016), https://doi.org/10.1051/cocv:1997102
  • [24] Lionel Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var. 10 (2004), no. 3, 346-380 (electronic). MR 2084328 (2005h:93091), https://doi.org/10.1051/cocv:2004012
  • [25] Lionel Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim. 39 (2000), no. 2, 331-351 (electronic). MR 1788062 (2001j:93012), https://doi.org/10.1137/S0363012999353229
  • [26] Lionel Rosier and Bing-Yu Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim. 45 (2006), no. 3, 927-956. MR 2247720 (2007h:35297), https://doi.org/10.1137/050631409
  • [27] Jean-Claude Saut and Nikolay Tzvetkov, On a model system for the oblique interaction of internal gravity waves, Special issue for R. Temam's 60th birthday, M2AN Math. Model. Numer. Anal. 34 (2000), no. 2, 501-523. MR 1765672 (2001f:35356), https://doi.org/10.1051/m2an:2000153
  • [28] Jacques Simon, Compact sets in the space $ L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65-96. MR 916688 (89c:46055), https://doi.org/10.1007/BF01762360
  • [29] Roger Temam, Navier-Stokes equations, 3rd ed., Theory and numerical analysis; With an appendix by F. Thomasset, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 769654 (86m:76003)
  • [30] O. P. Vera Villagran, Gain of regularity of the solutions of a coupled system of equations of Korteweg-de Vries type, Ph.D. Thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Brazil, (2001).
  • [31] Enrike 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), https://doi.org/10.1080/03605309908820684

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 93D15, 93B05, 35B40, 35Q53

Retrieve articles in all journals with MSC (2010): 93D15, 93B05, 35B40, 35Q53


Additional Information

A. F. Pazoto
Affiliation: Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil
Email: ademir@im.ufrj.br

G. R. Souza
Affiliation: Federal Center of Technology Education Celso Suckow da Fonseca, CEFET-RJ, Avenida Governador Roberto Silveira, 1900, CEP 28.635-000, Nova Friburgo, RJ, Brazil
Email: gilmar@im.ufrj.br

DOI: https://doi.org/10.1090/S0033-569X-2013-01343-1
Keywords: Exponential decay, stabilization, Korteweg-de Vries equation.
Received by editor(s): May 21, 2012
Published electronically: December 6, 2013
Additional Notes: The first author was partially supported by CNPq (Brazil).
The second author was partially supported by Capes (Brazil).
Article copyright: © Copyright 2013 Brown University

American Mathematical Society