A RESULT ON CLASS-C1 LINEARIZATION OF CONTRACTIONS IN INFINITE DIMENSIONS.

This note is a short presentation of our results in [5]. We start explaining, as a motivating example, a situation where a result of C1-linearization in infinite dimensions was needed and used. In the paper [2] it was proved that for some nonlinearities f(x, u) and some small values of α> 0 the global attractor of the dynamical system defined in H1(0, π) × L2(0, π) by the second order initial-boundary value problem utt + 2αut = uxx + f(x, u), 0 < x < π ux(0) = ux(π) = 0 is not contained on any finite-dimensional invariant manifold of class C1. In one of the steps of the proof it was needed to prove that for the case f(x, u) ≡ f(u) with f(0) = 0 and f ′(0) < 0 there are only countable many finite-dimensional invariant manifolds of class C1 containing the (assimptotically stable) equilibrium point (u, ut) = (0, 0). This fact was proved by linearization, that is by showing that under an abstract change of variables of class C1 in a neighborhood of (0, 0) ∈ H1(0, π)×L2(0, π), the equation turned into its linear part vtt + 2αvt = vxx + f ′(0)v, 0 < x < π vx(0) = vx(π) = 0