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Stability of periodic solutions to parabolic problems with nonlinear boundary conditions

We investigate non–autonomous quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results give principles of linearized (orbital) stability for solutions in the vicinity of a periodic solution. Our approach relies on a detailed study of regularity properties of the linearized problem.