Stable and Unstable Manifolds for Quasilinear Parabolic Problems with Fully Nonlinear Dynamical Boundary Conditions

We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper we treat center, center–stable and center–unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction–diffusion systems with dynamical boundary conditions and to the two–phase Stefan problem with surface tension.