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More results on the decay of solutions to nonlinear, dispersive wave equations. (English) Zbl 0870.35088

Summary: The asymptotic behaviour of solutions to the generalized regularized long-wave-Burgers equation \[ u_t+u_x+ u^pu_x-\nu u_{xx}- u_{xxt}=0\tag{1} \] is considered for \(\nu>0\) and \(p\geq 1\). Complementing recent studies which determined sharp decay rates for these kind of nonlinear, dispersive, dissipative wave equations, the present study concentrates on the more detailed aspects of the long-term structure of solutions. Scattering results are obtained which show enhanced decay of the difference between a solution of (1) and an associated linear problem. This in turn leads to explicit expressions for the large-time asymptotics of various norms of solutions of these equations for general initial data for \(p>1\) as well as for suitably restricted data \(p\geq 1\). Higher-order temporal asymptotics of solutions are also obtained. Our techniques may also be applied to the generalized Korteweg-de Vries-Burgers equation \[ u_t+ u_x+u^pu_x-\nu u_{xx}+u_{xxx}=0,\tag{2} \] and in this case our results overlap with those of Dix. The decay of solutions in the spatial variable \(x\) for both (1) and (2) is also considered.

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
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