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Decay rates of the solutions of nonlinear dispersive equations

Published online by Cambridge University Press:  14 November 2011

Vanilde Bisognin  and
Gustavo Perla Menzala
Vanilde Bisognin
Affiliation:
Department of Mathematics, Federal University of Santa Maria, RS, Santa Maria, Brazil
Gustavo Perla Menzala
Affiliation:
LNCC/CNPq, Rua Lauro Müller 455, Botafogo, CEP 22290, RJ, Brazil and IM-UFRJ, CP 68530, RJ, Brazil
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Abstract

We consider a family of dispersive equations whose simplest representative would be a Benjamin–Bona–Mahony equation with a Burger's type dissipation. The effect of possible unevenness of the bottom surface is considered and our main result gives decay rates of the solutions in Lβ(ℝ) spaces, 2 ≦ β ≦ + ∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1231 - 1246
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Abdelouhad, L.. Nonlocal dispersive equations in weighted Sobolev spaces. Differential Integral Equations 5 (2) (1992), 307–338.Google Scholar
2Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
3Amick, C. J., Bona, J. L. and Schonbek, M. E.. Decay of solutions of some nonlinear wave equations. J. Differential Equations 81 (1989), 1–49.CrossRefGoogle Scholar
4Bergh, J. and Lofstrom, J.. Interpolation Spaces–An Introduction (Berlin: Springer, 1976).CrossRefGoogle Scholar
5Biler, P.. Asymptotic behaviour in time of some equations generalising Korteweg de Vries-Burgers. Bull. Polish Acad. Sci. 32 (1984), m275–282.Google Scholar
6Bona, J. L. and Smith, R.. The initial value problem for the Korteweg de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555–604.Google Scholar
7Dix, D.. Temporal asymptotic behaviour of solutions of the Benjamin-Ono-Burgers equation. J. Differential Equations 81 (1991), 238–287.CrossRefGoogle Scholar
8Johnson, R. S.. On the development of a solitary wave moving over an uneven bottom. Proc. Cambridge Philos. Sci. 73 (1973), 183–203.CrossRefGoogle Scholar
9Kakutani, T.. Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30 (1) (1971), 272–276.CrossRefGoogle Scholar
10Racke, R.. Lectures on nonlinear evolution equations–Initial value problems. Aspects of Math. (Braunschweig: Vieweg, 1991).Google Scholar
11Souganidis, P. E. and Strauss, W. A.. Instability of a class of dispersive solitary waves. Proc. Roy. Soc. Edinburgh Sect A 114 (1990), 195–212.CrossRefGoogle Scholar
12Strauss, W. A.. Decay and asymptotics for ⋳u = F(u). J. Fund. Anal. 2 (1968), 409–457.CrossRefGoogle Scholar