Asymptotic behavior of solutions to the generalizedcubic double dispersion equation in one space dimension

Masakazu Kato; Yu-Zhu Wang; Shuichi Kawashima

doi:10.3934/krm.2013.6.969

Article Contents

2013, Volume 6, Issue 4: 969-987. Doi: 10.3934/krm.2013.6.969

This issue Previous Article Some properties of the kinetic equation for electron transport in semiconductors Next Article Remarks on the full dispersion Kadomtsev-Petviashvli equation

Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension

1.
Muroran Institute of Technology, Muroran 050-8585
2.
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011
3.
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395

Received: June 30, 2013

Revised: August 31, 2013

Published: November 2013

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Abstract

We study the initial value problem for the generalized cubic double dispersion equation in one space dimension. We establish a nonlinear approximation result to our global solutions that was obtained in [6]. Moreover, we show that as time tends to infinity, the solution approaches the superposition of nonlinear diffusion waves which are given explicitly in terms of the self-similar solution of the viscous Burgers equation. The proof is based on the semigroup argument combined with the analysis of wave decomposition.

Keywords:

Generalized cubic double dispersion equation,
global existence,
asymptotic behavior,
diffusion waves,
Burgers equation.

Mathematics Subject Classification: Primary: 35L30, 35L76, 35B40; Secondary: 35C06.

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