Largetime asymptotics of the generalized BenjaminOnoBurgers equation
Jerry L. Bona  Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States (email) Abstract: In this paper, attention is given to pure initialvalue problems for the generalized BenjaminOnoBurgers (BOB) equation $ u_t + u_x +(P(u))_{x}\nu $u_{xx}$  H$u_{xx}=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 largetime asymptotics of solutions of the initialvalue problem. If $\Lambda (s)$ is defined by $\Lambda '(s) = P(s), \Lambda (0) = 0,$ then solutions of the initialvalue 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 BenjaminOnoBurgers equation;
dissipation; dispersion; nonlinearity; global wellposedness; decay rates.
Received: March 2009; Revised: November 2009; Available Online: October 2010. 
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