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

Youcef Mammeri (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

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 \cite{bbm}, namely the equation $u_t + u_x +u^\rho u_x + H(u_{xt})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{xxt} +\partial_x^{-1}u_{yy} =0$. 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 $\frac{1}{2}$.

Mots clés: BO; BBM; KP equations; decay in time

Domaines: Mathématiques [math]

Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure and Applied Analysis, 2008, 7 (3), pp.513 - 532. ⟨10.3934/cpaa.2008.7.513⟩. ⟨hal-01090394⟩