Non-Linear Equations of Evolution

In the case of equation (2), it is well-known that if we assume that f satisfies a local Lipschitz condition in u, then local solutions of (2) with u(O) = uo will exist. (This is given in detail in [1] and [15]. A straightforward extension of this fact to the case of more general equations of evolution (1) with A(t) = A independent of t is carried through and exploited in [2], [13], and [26].) However, for infinite-dimensional spaces, the continuity hypothesis of Assumption (I) is not enough to assure the existence of local solutions. Our basic result for equation (2) is that Assumptions (I) and (II) above are sufficient for the existence and uniqueness of a strongly C1-function u from R+ to H which satisfies equation (2), and for which u(0) =u0. A similar result