A note on a non-local Kuramoto-Sivashinsky equation

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 -> infinity) parallel to u parallel to <= C L-nu from nu = 11/5 to nu = 3/2, and for the case of general initial data we establish an estimate of the above form with nu = 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).