Non-vanishing profiles for the Kuramoto–Sivashinsky equation on the infinite line

We study the Kuramoto–Sivashinsky equation on the infinite line with initial conditions having arbitrarily large limits ± Y at x = ± ∞. We show that the solutions have the same limits for all positive times. This implies that an attractor for this equation cannot be defined in L^∞. To prove this, we consider profiles with limits at x = ± ∞ and show that initial conditions L²-close to such profiles lead to solutions that remain L²-close to the profile for all times. Furthermore, the difference between these solutions and the initial profile tends to 0 as x → ± ∞, for any fixed time t > 0. Analogous results hold for L²-neighbourhoods of periodic stationary solutions. This implies that profiles and periodic stationary solutions partition the phase space into mutually unattainable regions.