Quasilinear Evolution Equations in [L.sup.P.sub.[mu]]-Spaces with Lower Regular Initial Data

We study the Cauchy problem of the quasilinear evolution equations in [L.sup.P.sub.[mu]]-spaces. Based on the theories of maximal [L.sup.p]-regularity of sectorial operators, interpolation spaces, and time-weighted [L.sup.p]-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in [L.sup.p,q]-spaces with temporal weights.