Quasilinear parabolic equations with superlinear nonlinearities in critical spaces

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established in a certain critical case of strict inclusion dom(f) dom(A) for the domains of the (superlinear) function u↦ f(u) and the quasilinear part u↦ A(u). 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.