Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation

Pages: 15 - 50, Volume 4, Issue 1, February 2011      doi:10.3934/dcdss.2011.4.15

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Jerry L. Bona - Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States (email)
Laihan Luo - Department of Mathematics, New York Institute of Technology, 1855 Broadway, New York, NY 10023, United States (email)

Abstract: In this paper, attention is given to pure initial-value problems for the generalized Benjamin-Ono-Burgers (BOB) equation

$ u_t + u_x +(P(u))_{x}-\nu $uxx$ - H$uxx=0,

where $H$ is the Hilbert transform, $\nu > 0$ and $P\ : R \to R$ is a smooth function. We study questions of global existence and of the large-time asymptotics of solutions of the initial-value problem. If $\Lambda (s)$ is defined by $\Lambda '(s) = P(s), \Lambda (0) = 0,$ then solutions of the initial-value problem corresponding to reasonable initial data maintain their integrity for all $t \geq 0$ provided that $\Lambda$ and $P'$ satisfy certain growth restrictions. In case a solution corresponding to initial data that is square integrable is global, it is straightforward to conclude it must decay to zero when $t$ becomes unboundedly large. We investigate the detailed asymptotics of this decay. For generic initial data and weak nonlinearity, it is demonstrated that the final decay is that of the linearized equation in which $P \equiv 0.$ However, if the initial data is drawn from more restricted classes that involve something akin to a condition of zero mean, then enhanced decay rates are established. These results extend the earlier work of Dix who considered the case where $P$ is a quadratic polynomial.

Keywords:  Generalized Benjamin-Ono-Burgers equation; dissipation; dispersion; nonlinearity; global well-posedness; decay rates.
Mathematics Subject Classification:  Primary: 35B40, 35C20, 35Q35; Secondary: 76D03, 76D09.

Received: March 2009;      Revised: November 2009;      Available Online: October 2010.