Decay of solutions to a water wave model with a nonlocal viscous dispersive term

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November 2010, 27(4): 1473-1492. doi: 10.3934/dcds.2010.27.1473

Decay of solutions to a water wave model with a nonlocal viscous dispersive term

Min Chen ^1, , S. Dumont ^2, , Louis Dupaigne ^3, and Olivier Goubet ^4,

1.	Department of Mathematics, Purdue University, West Lafayette, IN 47907
2.	LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex
3.	LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039Amiens Cedex 1
4.	Universite de Picardie Jules Verne, LAMFA UMR 7352, 33 rue Saint-Leu, 80039 Amiens cedex

Received July 2009 Revised January 2010 Published March 2010

Abstract
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In this article, we investigate a water wave model with a nonlocal viscous term

$ u_t+u_x+\beta $u_xxx$+\frac{\sqrt{\nu}}{\sqrt{\pi}}\int_0^t \frac{u_t(s)}{\sqrt{t-s}}ds+$uu_x$=\nu $u_xx$. $

The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

Keywords: long-time asymptotics., nonlocal, viscous asymptotic models, Nonlinear water waves.

Mathematics Subject Classification: Primary: 35Q35, 35Q53, 76B15; Secondary: 65M7.

Citation: Min Chen, S. Dumont, Louis Dupaigne, Olivier Goubet. Decay of solutions to a water wave model with a nonlocal viscous dispersive term. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1473-1492. doi: 10.3934/dcds.2010.27.1473

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