  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.
