On the decay in time of solutions of some generalized regularized long waves equations

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May 2008, 7(3): 513-532. doi: 10.3934/cpaa.2008.7.513

On the decay in time of solutions of some generalized regularized long waves equations

1.	Laboratoire Paul Painlevé, Université des Sciences et Technologies Lille 1, 59 655 Villeneuve d’Ascq, France

Received April 2007 Revised October 2007 Published February 2008

Abstract
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We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation $u_t + u_x +u^\rho u_x + H(u_{x t})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{x x t} +\partial_x^{-1}u_{y y} =0$. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power $\rho$ of the nonlinearity is higher than $3$, the respective solution of these equations tends to zero when time rises with a decay rate of order close to $\frac{1}{2}$.

Keywords: KP equations, decay in time., BBM, BO.

Mathematics Subject Classification: 35B05, 35B40, 35Q53, 76B03, 76B5.

Citation: Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 513-532. doi: 10.3934/cpaa.2008.7.513

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