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July  2008, 7(4): 867-881. doi: 10.3934/cpaa.2008.7.867

Sharp well-posedness results for the Kuramoto-Velarde equation

Didier Pilod 1,

1. 

UFRJ, Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil

Received  June 2007 Revised  January 2008 Published  April 2008

We study the dispersive Kuramoto-Sivashinsky and Kuramoto-Velarde equations. We show that the associated initial value problem is locally (and globally in some cases) well-posed in Sobolev spaces $H^s(\mathbb R)$ for $s > -1$. We also prove that these results are sharp in the sense that the flow map of these equations fails to be $C^2$ in $H^s(\mathbb R)$ for $s < -1$. In addition, we determine the limiting behavior of the solutions when the dispersive parameter tends to zero.
Keywords: initial value problem., Nonlinear PDE.
Mathematics Subject Classification: Primary: 35Q53, 35K55; Secondary: 76B03, 76B1.
Citation: Didier Pilod. Sharp well-posedness results for the Kuramoto-Velarde equation. Communications on Pure and Applied Analysis, 2008, 7 (4) : 867-881. doi: 10.3934/cpaa.2008.7.867
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