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We study quasilinear systems of parabolic partial differential equa- tions with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicin- ity of an equilibrium. The local center, center-stable, and center-unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.

Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions