Analyticity of Stable Invariant Manifolds for Ginzburg-landau Equation

This paper is devoted to prove analyticity of stable invariant manifold in a neighbourhood of an unstable steady-state solution for GinzburgLandau equation defined in a bounded domain of dimension not more than three. This investigation is made for possible applications in stabilization theory for semilinear parabolic equation. Introduction In this paper we prove analyticity of stable invariant manifoldM− near unstable steady-state solution of Ginzburg-Landau equation. This result can be used in stabilization theory for semilinear parabolic PDE defined in a bounded domain Ω with feedback Dirichlet control given on the boundary ∂Ω or on its open part. This theory for general quasilinear parabolic equation and for Navier-Stokes system was built in [F1], [F2], [F3]. We have to emphasize that the main reason to develop stabilization theory is to provide reliable stable algorithms for numerical stabilization. To construct such algorithms it is very desirable to have a simple description for infinite-dimensional invariant manifold M− allowing to calculate it easily in arbitrary point. Just such description gives functional-analytic decomposition of M−. Using classical description of M− by means of a map F (y−) (see [BV], [Hen]), one can look for this map as a serie