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.