Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 151 - 193, Volume 1, Issue 2, April 1995      doi:10.3934/dcds.1995.1.151

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Jerry L. Bona - Department of Mathematics and Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802, United States (email)
Laihan Luo - Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, United Kingdom (email)

Abstract: 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.

Keywords:  Nonlinear, dispersive, dissipative, wave equations; Korteweg-de Vries-Burgers equation; regularized long-wave-Burgers equation; decay rates; large-time asymptotics.
Mathematics Subject Classification:  35Q53.

Received: November 1994;      Available Online: February 1995.