Preprint series: 06-02, Reports on Analysis

MSC:

35B40 Asymptotic behavior of solutions

35B65 Smoothness/regularity of solutions of PDE

35K35 Boundary value problems for higher-order, parabolic equations

35K50 Boundary value problems for parabolic systems

35K57 Reaction-diffusion equations

Abstract: We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev-Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed.

Keywords: Local wellposedness, regularity, linearized stability, hyperbolic equilibrium, invariant manifold, maximal regularity, anisotropic Slobodetskii spaces, Nemytskii operators, exponential dichotomy, extrapolation, implicit function theorem, reaction diffusion equation.

Upload: 2006-02-22

Update: 2006 -02 -22