April  1995, 1(2): 151-193. doi: 10.3934/dcds.1995.1.151

More results on the decay of solutions to nonlinear, dispersive wave equations

1. 

Department of Mathematics and Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802, United States

2. 

Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, United Kingdom

Received  November 1994 Published  February 1995

The asymptotic behaviour of solutions to the generalized regularized long-wave-Burgers equation

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}-u_{x xt}=0$ ($*$)

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 ($*$) 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 for $p \geq 1.$ Higher-order temporal asymptotics of solutions are also obtained. Our techniques may also be applied to the generalized Korteweg-de Vries-Burgers equation

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}+u_{x x x}=0,$ ($**$)

and in this case our results overlap with those of Dix. The decay of solutions in the spatial variable $x$ for both ($*$) and ($**$) is also considered.

Citation: Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151
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