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