Well-Posedness of Quasilinear Parabolic Equations in Time-Weighted Spaces

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u^{\mathrm{\prime }}=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{d}\mathrm{o}\mathrm{m}(f)⊊\mathrm{d}\mathrm{o}\mathrm{m}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.