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 & 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.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[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.  doi: 10.1017/S0308210500021478.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.1007/s00245-011-9156-7.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.08.003.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1137/0149070.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1016/j.jde.2008.11.002.  Google Scholar

[16]

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

[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.   Google Scholar

[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, (1994).   Google Scholar

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

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.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[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.  doi: 10.1017/S0308210500021478.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.1007/s00245-011-9156-7.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.08.003.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1137/0149070.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.1016/j.jde.2008.11.002.  Google Scholar

[16]

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

[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.   Google Scholar

[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, (1994).   Google Scholar

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1103/PhysRevLett.15.240.  Google Scholar

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