The time dependent amplitude equation for the Swift-Hohenberg problem

Precise estimates for the validity of the amplitude approximation for the swift-Hohenberg equation are given, in a fully time dependent framework. It is shown that small solutions of orderO(ɛ) which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of orderO(ɛ-2). For the proofs, we use techniques for nonlinear semigroups and oscillatory integrals.