# American Institute of Mathematical Sciences

November  2007, 18(4): 701-707. doi: 10.3934/dcds.2007.18.701

## 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, United States 2 Department of Mathematics, Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6, Canada 3 Department of Computer Science, University of Illinois Urbana-Champaign, 1409 W. Green St, Urbana IL 61801, United States

Received  September 2006 Revised  January 2007 Published  May 2007

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 $O(1)$.
Citation: Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
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