Strong periodic solutions to quasilinear parabolic equations: An approach by the Da Prato–Grisvard theorem

Abstract

This article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato-Grisvard theorem on maximal 𝐿 𝑝 -regularity in real interpolation spaces. The method is used to show that quasilinear Keller-Segel systems admit a unique, strong 𝑇-periodic solution in a neighborhood of 0 provided the external forces are 𝑇-periodic and satisfy certain smallness conditions. A similar assertion applies to a Nernst-Planck-Poisson type system in electrochemistry. The proof for the quasilinear Keller-Segel systems relies also on a new mixed derivative theorem in real interpolation spaces, that is, Besov spaces, which is of independent interest.