Existence of solutions of some nonlinear wave equations (Q1063183)

The author considers the initial value problem stated as follows: \[ d^ 2u/dt^ 2+Au+\partial \phi u\ni f(t,u),\quad t\in | 0,T|,\quad u\in H,\quad u(0)=a,\quad (d/dt)u(0)=b. \] Here t is a fixed positive number and H is a real Hilbert space while A is a positive selfadjoint operator in H, \(\phi\) is a known function from H to [-\(\infty,\infty]\), f is a given function from [0,T]\(\times H\) to H and \(\partial \phi\) is the subdifferential of \(\phi\). The main objective of the paper is to give some sufficient conditions which guarantee the existence of at least one solution. Though the paper involves some illustrative examples, it is hardly readable. Especially the meanings of some relations appearing in the definition of a solution are not clear.