Viability for Nonautonomous Semilinear Differential Equations

Let X be a reflexive Banach space, A: D(A) X→X the infinitesimal generator of a compact C0-semigroup S(t): X→X, t 0, D a locally closed subset in X and (t, x) f(t, x) a function on [a, b)×D which is locally integrably bounded, measurable with respect to t and continuous with respect to x. The main result of the paper is: T . Under the general assumptions above a necessary and sufficient condition in order that for each (τ, ξ) [a, b)×Dthere exists at least one mild solutionu: [τ, T]→Dof satisfyingu(τ)=ξis the tangency condition below: There is a negligible subset Z of [a, b) such that for eacht [a, b)\Zand eachx D,