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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

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

$ u_t+u_x+\beta $uxxx$+\frac{\sqrt{\nu}}{\sqrt{\pi}}\int_0^t \frac{u_t(s)}{\sqrt{t-s}}ds+$uux$=\nu $uxx$. $

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 and Continuous Dynamical Systems, 2010, 27 (4) : 1473-1492. doi: 10.3934/dcds.2010.27.1473
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