On the decay in time of solutions of some generalized regularized long waves equations - Archive ouverte HAL Access content directly
Journal Articles Communications on Pure and Applied Analysis Year : 2008

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

Abstract

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 $\frac{1}{2}$.
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Dates and versions

hal-01090394 , version 1 (03-12-2014)

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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⟩
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