  The real Ginzburg-Landau equation arises as a universal amplitude equation
for the description of pattern-forming systems exhibiting a Turing bifurcation.
It possesses spatially periodic roll solutions which are known to be stable
against localized perturbations. It is the purpose of this paper to prove their
stability against bounded perturbations, which are not necessarily localized.
Since all state-of-the-art techniques rely on localization or periodicity
properties of perturbations, we develop a new method, which employs pure
$L^\infty$-estimates only. By fully exploiting the smoothing properties of the
semigroup generated by the linearization, we are able to close the nonlinear
iteration despite the slower decay rates. To show the wider relevance of our
method, we also apply it to the amplitude equation as it appears for
pattern-forming systems with an additional conservation law.
