# American Institute of Mathematical Sciences

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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