# American Institute of Mathematical Sciences

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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