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 Free Access
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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
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- 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, DOI https://doi.org/10.1016/S0362-546X%2897%2900724-4
- 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
- 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, DOI https://doi.org/10.1137/0514085
- 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
- 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, DOI https://doi.org/10.1007/s00245-011-9156-7
- 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, DOI https://doi.org/10.1142/S021919971100418X
- 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).
- 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, DOI https://doi.org/10.1002/sapm1984703235
- 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, DOI https://doi.org/10.1080/03605300903585336
- 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, DOI https://doi.org/10.3934/cpaa.2004.3.417
- 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, DOI https://doi.org/10.1016/j.jde.2008.11.002
- 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. MR 2276662, DOI https://doi.org/10.1090/S0002-9939-07-08810-7
- 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
- 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, DOI https://doi.org/10.1090/S0033-569X-2011-01245-6
- 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, DOI https://doi.org/10.1002/mma.847
- 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, DOI https://doi.org/10.1090/qam/1878262
- 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
- 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, DOI https://doi.org/10.1142/S0219199709003600
- L. Molinet and D. Pilod, The Cauchy problem for the Benjamin-On equation in $L^2$ revisited, Analysis and PDE, to appear.
- 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, DOI https://doi.org/10.3934/mcrf.2011.1.353
- 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. MR 2148854, DOI https://doi.org/10.1051/cocv%3A2005015
- Lionel Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33–55. MR 1440078, DOI https://doi.org/10.1051/cocv%3A1997102
- Lionel Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var. 10 (2004), no. 3, 346–380. MR 2084328, DOI https://doi.org/10.1051/cocv%3A2004012
- 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. MR 1788062, DOI https://doi.org/10.1137/S0363012999353229
- 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, DOI https://doi.org/10.1137/050631409
- Jean-Claude Saut and Nikolay Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal. 34 (2000), no. 2, 501–523. Special issue for R. Temam’s 60th birthday. MR 1765672, DOI https://doi.org/10.1051/m2an%3A2000153
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
- 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).
- 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, DOI https://doi.org/10.1080/03605309908820684
References
- 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)
- 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), DOI https://doi.org/10.1016/S0362-546X%2897%2900724-4
- 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)
- 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), DOI https://doi.org/10.1137/0514085
- 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)
- 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, DOI https://doi.org/10.1007/s00245-011-9156-7
- 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), DOI https://doi.org/10.1142/S021919971100418X
- 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).
- 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)
- 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), DOI https://doi.org/10.1080/03605300903585336
- 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), DOI https://doi.org/10.3934/cpaa.2004.3.417
- 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), DOI https://doi.org/10.1016/j.jde.2008.11.002
- 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), DOI https://doi.org/10.1090/S0002-9939-07-08810-7
- 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)
- 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), DOI https://doi.org/10.1090/S0033-569X-2011-01245-6
- 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), DOI https://doi.org/10.1002/mma.847
- 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)
- 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
- 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), DOI https://doi.org/10.1142/S0219199709003600
- L. Molinet and D. Pilod, The Cauchy problem for the Benjamin-On equation in $L^2$ revisited, Analysis and PDE, to appear.
- 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), DOI https://doi.org/10.3934/mcrf.2011.1.353
- 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), DOI https://doi.org/10.1051/cocv%3A2005015
- 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), DOI https://doi.org/10.1051/cocv%3A1997102
- 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), DOI https://doi.org/10.1051/cocv%3A2004012
- 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), DOI https://doi.org/10.1137/S0363012999353229
- 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), DOI https://doi.org/10.1137/050631409
- 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), DOI https://doi.org/10.1051/m2an%3A2000153
- 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), DOI https://doi.org/10.1007/BF01762360
- 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)
- 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).
- 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), DOI https://doi.org/10.1080/03605309908820684
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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
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