Non-Vanishing Profiles for the Kuramoto-Sivashinsky Equation on the Infinite Line

Abstract

We study the Kuramoto-Sivashinsky equation on the infinite line with initial conditions having arbitrarily large limits $\pm Y$ at $x=\pm \infty$. 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 $L^{\infty }$. To prove this, we consider profiles with limits at $x=\pm \infty$, and show that initial conditions $L^2$-close to such profiles lead to solutions which remain $L^2$-close to the profile for all times. Furthermore, the difference between these solutions and the initial profile tends to 0 as $x\to \pm \infty$, for any fixed time $t>0$. Analogous results hold for $L^2$-neighborhoods of periodic stationary solutions. This implies that profiles and periodic stationary solutions partition the phase space into mutually unattainable regions.