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

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 $\frac{1}{2}$.
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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