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Decay of solutions to nonlinear, dispersive wave equations. (English) Zbl 0780.35098

Summary: The asymptotic behavior of solutions to the initial-value problem for the generalized Korteweg-de Vries-Burgers equation \[ u_ t+u_ x+u^ p u_ x- \nu u_{xx}+u_{xxx}=0 \] and the generalized regularized long- wave-Burgers equation \[ u_ t+u_ x+u^ p u_ x- \nu u_{xx}- u_{xxt}=0 \] is studied for \(\nu>0\) and \(p\geq 2\). The decay rate of solutions of these equations is that exhibited by solutions of the linearized equation in which the nonlinear term is simply dropped.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35C15 Integral representations of solutions to PDEs
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