January  2016, 21(1): 245-252. doi: 10.3934/dcdsb.2016.21.245

Long-time behavior of solutions of the generalized Korteweg--de Vries equation

1. 

ALHOSN University, Mathematics and Natural Sciences Department, PO Box 38772, Abu Dhabi

Received  August 2014 Revised  August 2015 Published  November 2015

In this paper, we study the large-time behavior of solutions to the initial-value problem for the generalized Korteweg--de Vries equation. We show that for initial data in some weighted space, the asymptotic behavior of the solution can be improved. In addition, we give the asymptotic profile of the fundamental solution of the linearized model. We extend and improve the results in [3] and [2].
Citation: Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.

[2]

J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502. doi: 10.1017/S0308210500021478.

[3]

J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations, 6 (1993), 961-980.

[4]

G. Bowtell and A. E. G Stuart, A particle representation for Korteweg-de Vries solitons, J. Math. Phys., 24 (1983), 969-981. doi: 10.1063/1.525786.

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, A. Faminskii and F. Natal, Decay of solutions to damped Korteweg-de Vries type equation, Appl. Math. Optim., 65 (2002), 221-251. doi: 10.1007/s00245-011-9156-7.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, V. Komornik and J. H. Rodrigues, Global well-posedness and exponential decay rates for a KdV-Burgers equation with indefinite damping, Ann. I. H. Poincaré, 31 (2014), 1079-1100. doi: 10.1016/j.anihpc.2013.08.003.

[7]

C. Guo and S. Fang, Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation, Nonlinear Anal., 75 (2012), 3385-3392. doi: 10.1016/j.na.2011.12.035.

[8]

N. Hayashi and P. I. Naumkin, Asymptotics for the Korteweg-de Vries-Burgers equation, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1441-1456. doi: 10.1007/s10114-005-0677-3.

[9]

P. F. Hodnett and T. P. Moloney, On the structure during interaction of the two-soliton solution of the Korteweg-de Vries equation, SIAM J. Appl. Math., 49 (1989), 1174-1187. doi: 10.1137/0149070.

[10]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.

[11]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Rev., 14 (1972), 582-643. doi: 10.1137/1014101.

[12]

G. Karch, Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal., 35 (1999), 199-219.

[13]

C. J. Knickerbocker and A. C. Newell, Shelves and the Korteweg-de Vries equation, Journal of Fluid Mechanics, 98 (1980), 803-818. doi: 10.1017/S0022112080000407.

[14]

P. D Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.

[15]

F. Linares and A. F. Pazoto, Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane, J. Differential Equations, 246 (2009), 1342-1353. doi: 10.1016/j.jde.2008.11.002.

[16]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. Kyoto. Univ, 12 (1976), 169-189. doi: 10.2977/prims/1195190962.

[17]

P. I. Naumkin, On the asymptotic behavior for large time values of the solutions of nonlinear equations in the case of maximal order, Diff. Equations,, 29 (1993), 1071-1074.

[18]

P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, Volume 133 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by Boris Gommerstadt.

[19]

R. Racke, Lectures on Nonlinear Evolution Equations. Initial value Problems. Aspects of Mathematics, E19, Friedrich Vieweg and Sohn: Braunschweig, Wiesbaden, 1992. doi: 10.1007/978-3-663-10629-6.

[20]

I. E. Segal, Dispersion for non-linear relativistic equations, II, Ann. Sci. Ecole Norm. Sup., 1 (1968), 459-497.

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptotic Analysis, 68 (2010), 155-186. doi: 10.3233/ASY-2010-0988.

[22]

N. J Zabusky and M. D Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett, 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

show all references

References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.

[2]

J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502. doi: 10.1017/S0308210500021478.

[3]

J. L. Bona and L. Luo, Decay of solutions to nonlinear, dispersive wave equations, Differential Integral Equations, 6 (1993), 961-980.

[4]

G. Bowtell and A. E. G Stuart, A particle representation for Korteweg-de Vries solitons, J. Math. Phys., 24 (1983), 969-981. doi: 10.1063/1.525786.

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, A. Faminskii and F. Natal, Decay of solutions to damped Korteweg-de Vries type equation, Appl. Math. Optim., 65 (2002), 221-251. doi: 10.1007/s00245-011-9156-7.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, V. Komornik and J. H. Rodrigues, Global well-posedness and exponential decay rates for a KdV-Burgers equation with indefinite damping, Ann. I. H. Poincaré, 31 (2014), 1079-1100. doi: 10.1016/j.anihpc.2013.08.003.

[7]

C. Guo and S. Fang, Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation, Nonlinear Anal., 75 (2012), 3385-3392. doi: 10.1016/j.na.2011.12.035.

[8]

N. Hayashi and P. I. Naumkin, Asymptotics for the Korteweg-de Vries-Burgers equation, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1441-1456. doi: 10.1007/s10114-005-0677-3.

[9]

P. F. Hodnett and T. P. Moloney, On the structure during interaction of the two-soliton solution of the Korteweg-de Vries equation, SIAM J. Appl. Math., 49 (1989), 1174-1187. doi: 10.1137/0149070.

[10]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.

[11]

A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Rev., 14 (1972), 582-643. doi: 10.1137/1014101.

[12]

G. Karch, Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal., 35 (1999), 199-219.

[13]

C. J. Knickerbocker and A. C. Newell, Shelves and the Korteweg-de Vries equation, Journal of Fluid Mechanics, 98 (1980), 803-818. doi: 10.1017/S0022112080000407.

[14]

P. D Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.

[15]

F. Linares and A. F. Pazoto, Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane, J. Differential Equations, 246 (2009), 1342-1353. doi: 10.1016/j.jde.2008.11.002.

[16]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. Kyoto. Univ, 12 (1976), 169-189. doi: 10.2977/prims/1195190962.

[17]

P. I. Naumkin, On the asymptotic behavior for large time values of the solutions of nonlinear equations in the case of maximal order, Diff. Equations,, 29 (1993), 1071-1074.

[18]

P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, Volume 133 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by Boris Gommerstadt.

[19]

R. Racke, Lectures on Nonlinear Evolution Equations. Initial value Problems. Aspects of Mathematics, E19, Friedrich Vieweg and Sohn: Braunschweig, Wiesbaden, 1992. doi: 10.1007/978-3-663-10629-6.

[20]

I. E. Segal, Dispersion for non-linear relativistic equations, II, Ann. Sci. Ecole Norm. Sup., 1 (1968), 459-497.

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations, Asymptotic Analysis, 68 (2010), 155-186. doi: 10.3233/ASY-2010-0988.

[22]

N. J Zabusky and M. D Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett, 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240.

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