Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models

We first show asymptotic L bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions. We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We establish rigorously the general structure of the spectrum of the linearized operator. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the nonresonant regime L 6= nπ, we prove conditional asymptotic stability, provided one considers only mean value zero data. If however L = n0π (but still ∫ L −L u0(x)dx = 0), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero) function as long as the initial data is small and lies on a center-stable manifold of co-dimension 2(n0 − 1).