A note on a non-local Kuramoto-Sivashinsky equation
Jared C. Bronski Razvan C. Fetecau Thomas N. Gambill
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 $O(1)$.
keywords: global attractors. Kuramoto-Sivashinsky equation

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