Abstract Cauchy problem for weakly continuous operators

We discuss the solvability of the abstract Cauchy problem u′(t)=Au(t)u′(t)=Au(t) for t>0t>0 and u(0)=u0∈Du(0)=u0∈D, where D is a weakly closed subset of a Banach space Y continuously embedded in the underlying Banach space X and A is a nonlinear operator from D into X satisfying a local weak continuity condition described by a family of functionals. We introduce a new type of weak compactness on the level sets of functionals and give a necessary and sufficient condition for the existence of a weakly continuously differentiable solution satisfying a growth condition. We apply the abstract result obtained to the Cauchy problem for a nonlinear Schrödinger equation and a mixed problem for a logarithmic wave equation.