OPTIMAL PARAMETER-DEPENDENT BOUNDS FOR KURAMOTO-SIVASHINSKY-TYPE EQUATIONS

We derive a priori estimates on the absorbing ball in L-2 for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Non linearity 19, 2023-2039 (2006)) to take into account the dependence on both large and small parameters in the system. In the case of the destabilized KS equation, the rigorous bound lim sup(t ->infinity) parallel to u parallel to <= K alpha L-3/2 is sharp in both the large parameter a and the system size L. We also apply our methods to improve previous estimates on a nonlocal variant of the KS equation.