Quasilinear degenerate evolution equations in Banach spaces

The quasilinear degenerate evolution equation of parabolic type 0< t T considered in a Banach space X is written, putting Mv = u, in the from 0< t T, where A(u)=L(u)M^–1 are multivalued linear operators in X for u K, K being a bounded ball ||u||_Z<R in another Banach space Z continuously embedded in X. Existence and uniqueness of the local solution for the related Cauchy problem are given. The results are applied to quasilinear elliptic-parabolic equations and systems.