Semilinear Periodic-Parabolic Equations with Nonlinear Boundary Conditions

Recently much work has been devoted to periodic-parabolic equations with linear homogeneous boundary conditions. However, very little has been accomplished in the literature for periodic-parabolic problems with nonlinear boundary conditions. It is the purpose of this paper to prove existence and regularity results for (classical) periodic solutions to semilinear second order parabolic partial differential equations with nonlinear boundary conditions provided ordered upper and lower solutions are given. Fractional order function spaces, Ehrling–Gagliardo–Nirenberg and Lions–Peetre–Calderón type interpolation inequalities for functions in (anisotropic) Sobolev–Slobodeckii spaces play an important role in the obtainment ofa prioriboundary and interior estimates. In proving our existence results we make use of topological degree techniques and regularity results for linear parabolic partial differential equations under linear nonhomogeneous boundary conditions. We also indicate how one can obtain minimal and maximal time-periodic solutions to parabolic problems with nonlinear boundary conditions.