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Decay of solutions of a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system

Published online by Cambridge University Press:  14 November 2011

Zhang Linghai
Affiliation:
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, People's Republic of China

Abstract

We study decay estimates for the solutions to the initial value problem for a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. The method is integral estimation.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Albert, John. Dispersion of low energy waves for the generalized Benjamin–Bona–Mahony Equation. J. Differential Equations 63 (1986), 117134.CrossRefGoogle Scholar
2Amick, C. J., Bona, J. L. and Schonbek, M. E.. Decay of solutions of some nonlinear wave equations. J. Differential Equations 81 (1989), 149.CrossRefGoogle Scholar
3Biler, P.. Asymptotic behavior in time of solutions to some equations generalizing the Korteweg–de Vries–Burgers equation. Bull. Polish Acad. Sci. Math. 32 (1984), 275281.Google Scholar
4Biler, P.. Long time behavior of solutions of the generalized Benjamin–Bona–Mahony equation in two space dimensions. Differential and Integral Equations 5(4) (1992), 891901.CrossRefGoogle Scholar
5Bona, J. L. and Smith, R.. The initial value problem for the Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555601.Google Scholar
6Dix, Daniel B.. Temporal asymptotic behavior of solutions of the Benjamin–Ono–Burgers equations. J. Differential Equations 90 (1991), 238287.Google Scholar
7Boling, Guo. A class of general Korteweg–de Vries equation. Acta Math. Sinica 25 (1982), 641656.Google Scholar
8Pingfan, He. Global solutions for a coupled Korteweg–de Vries system. J. Partial Differential Equations 2(1989), 1630.Google Scholar
9Hopf, E.. The partial differential equation U t + UU x = μU xx. Comm. Pure Appl. Math. 3 (1950), 201230.Google Scholar
10Kato, T.. On the Korteweg–de Vries equation. Manuscripta Math. 28 (1979), 8999.CrossRefGoogle Scholar
11Kruskal, M. D., Miura, R. M. and Gardner, C. S.. Korteweg–de Vries equation and generalizations, Uniqueness, V. and nonexistence of polynomial conservation laws. J. Math. Phys. 11 (1970), 952.CrossRefGoogle Scholar
12Chenxia, Miao. Initial boundary value problem for the Korteweg–de Vries equation (MS thesis, Institute of Applied Physics and Computational Mathematics, 1990).Google Scholar
13Miura, R. M., Gardner, C. S. and Kruskal, M. D.. Korteweg–de Vries equation and generalizations, II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 12041209.CrossRefGoogle Scholar
14Pego, R. L.. Stability in systems of conservation laws with dissipation. Lectures in Appl. Math. 23 (1986), 345357.Google Scholar
15Schonbek, M. E.. Decay of solutions to parabolic conservation laws. Comm. Partial Differential Equations 7 (1980), 449473.CrossRefGoogle Scholar
16Schonbek, M. E.. L2 decay for weak solutions of the nonlinear Navier–Stokes equations. Arch. Rational Mech. Anal. 88 (1985), 209222.CrossRefGoogle Scholar
17Sjoberg, A.. On the Korteweg–de Vries equation, existence and uniqueness. J. Math. Anal. Appl. 29 (1970), 569579.CrossRefGoogle Scholar
18Strauss, W. A.. Dispersion of low energy waves for two conservative equations. Arch. Rational Mech. Anal. 55 (1974), 86.CrossRefGoogle Scholar
19Whitham, B.. Nonlinear dispersive waves. Proc. Roy. Soc. London Ser. A 283 (1965), 238261.Google Scholar
20Linghai, Zhang. Initial value problem for a nonlinear parabolic equation with singular integral-differential term. Acta Math. Appl. Sinica 8(4) (1992), 367376.Google Scholar
21Linghai, Zhang. Initial value problem for a nonlinear evolution system with singular integral-differential terms. J. Partial Differential Equations 7(1) (1994).Google Scholar
22Linghai, Zhang. Decay estimates for the solutions of some nonlinear evolution equations. J. Differential Equations (to appear 115 (1995)).Google Scholar
23Yulin, Zhou and Boling, Guo. The periodic boundary value problems and the initial value problems for the generalized Korteweg–de Vries systems of higher order. Acta Math. Sinica 27 (1984), 154176.Google Scholar