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2007, Volume 18, Issue 4: 701-707. Doi: 10.3934/dcds.2007.18.701
This issue Previous Article Mild mixing property for special flows under piecewise constant functions Next Article Existence of a semilinear elliptic system with exponential nonlinearities

A note on a non-local Kuramoto-Sivashinsky equation

  • 1.

    Department of Mathematics, University of Illinois Urbana-Champaign, 1409 W. Green St, Urbana IL 61801

  • 2.

    Department of Mathematics, Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6

  • 3.

    Department of Computer Science, University of Illinois Urbana-Champaign, 1409 W. Green St, Urbana IL 61801

Received:  August 31, 2006
Revised:  December 31, 2006
Published:  September 2007
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    Abstract
  • In this note we outline some improvements to a result of Hilhorst, Peletier, Rotariu and Sivashinsky [5] on the $L_2$ boundedness of solutions to a non-local variant of the Kuramoto-Sivashinsky equation with additional stabilizing and destabilizing terms. We are able to make the following improvements: in the case of odd data we reduce the exponent in the estimate lim sup$_t\rightarrow \infty$ ||$u$ || $\le C L^{\nu}$ from $\nu = \frac{11}{5}$ to $\nu=\frac{3}{2}$, and for the case of general initial data we establish an estimate of the above form with $\nu = \frac{13}{6}$. We also remove the restrictions on the magnitudes of the parameters in the model and track the dependence of our estimates on these parameters, assuming they are at least $O(1)$.
    Mathematics Subject Classification: 35B40, 35K30.

    Citation:
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