Stability of periodic solutions to parabolic problems with nonlinear boundary conditions. (English) Zbl 1260.35008

Summary: We investigate nonautonomous quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions. We establish local well-posedness and study the time and space regularity of the solutions. Our main results give principles of linearized (orbital) stability and instability for solutions in the vicinity of a periodic solution. Our approach relies on a detailed study of regularity properties of the linearized nonautonomous problem and its evolution family.


35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K59 Quasilinear parabolic equations
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B10 Periodic solutions to PDEs