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 and Continuous Dynamical Systems, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
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