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Traveling waves and weak solutions for an equation with degenerate dispersion


Authors: David M. Ambrose and J. Douglas Wright
Journal: Proc. Amer. Math. Soc. 141 (2013), 3825-3838
MSC (2010): Primary 35Q53
DOI: https://doi.org/10.1090/S0002-9939-2013-12070-8
Published electronically: July 11, 2013
MathSciNet review: 3091772
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Abstract: We consider the following family of equations:

$\displaystyle u_t = 2 u u_{xxx} - u_x u_{xx} + 2 k u u_x.$    

Here $ k \ne 0$ is a constant and $ x \in [-L_0,L_0]$. We demonstrate that for these equations there are compactly supported traveling wave solutions (which are in $ H^2$), and the Cauchy problem (with $ H^2$ initial data) possesses a weak solution which exists locally in time. These are the first degenerate dispersive evolution PDE where both of these features are known to hold simultaneously. Moreover, if $ k < 0$ or $ L_0$ is not too large, the solution exists globally in time.

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Additional Information

David M. Ambrose
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: ambrose@math.drexel.edu

J. Douglas Wright
Affiliation: Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email: jdoug@math.drexel.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-12070-8
Received by editor(s): January 9, 2012
Published electronically: July 11, 2013
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.