On Second-Order Differential Equations with Nonsmooth Second Member

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f in (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases of Fu. In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).