Quasilinear parabolic equations with superlinear nonlinearities in critical spaces

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations $u^{\prime }=A\left(u\right)u+f\left(u\right)$ is established in a certain critical case of strict inclusion $\mathrm{dom}\left(f\right)⊊\mathrm{dom}\left(A\right)$ for the domains of the (superlinear) function $u\mapsto f\left(u\right)$ and the quasilinear part $u\mapsto A\left(u\right)$. Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space. The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.