Abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators

In this paper we study the abstract Cauchy problem for quasi-linear evolution equation u'(t) = A(u(t))u(t), where {A(w); w is an element of W} is a family of closed linear operators in a real Banach space X such that D(A(w)) = Y for w is an element of W, and W is an open subset of another Banach space Y which is continuously embedded in X. The purpose of this paper is not only to establish a 'global' well-posedness theorem without assuming that Y is dense in X but also to propose a new type of dissipativity condition which is closely related with the continuous dependence of solutions on initial data.