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

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 Lp$L^p$‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong T$T$‐periodic solution in a neighborhood of 0 provided the external forces are T$T$‐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. [ABSTRACT FROM AUTHOR]