Abstract Cauchy problems for quasilinear operators whose domains are not necessarily dense or constant

The solvability of the abstract Cauchy problem for the quasilinear evolution equation u′(t)=A(u(t))u(t) for t>0 and u(0)=u0∈D is discussed. Here {A(w);w∈Y} is a family of closed linear operators in a real Banach space X such that Y⊂D(A(w))⊂Y¯ for w∈Y, Y is another Banach space which is continuously embedded in X, and D is a closed subset of Y. The existence and uniqueness of C1 solutions to the Cauchy problem is proved without assuming that Y is dense in X or D(A(w)) is independent of w. The abstract result is applied to obtain an L1-valued C1-solution to a size-structured population model.