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.
"Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions." Adv. Differential Equations 22 (7/8) 541 - 592, July/August 2017.