On certain nonlinear differential equations of second order in time (Q1078761)

The author considers initial-value problems associated with the following equation in a Hilbert space \(H:\) \((d^ 2/dt^ 2)u(t) + \partial\phi(u(t)) + \partial I_ K(u(t)) \ni f(t,u(t))\) where \(\phi\) is a coercive proper convex l.s.c. functional on a dense subspace of H; the sub-differential \(\partial \phi\) is assumed single valued and \(I_ K\) is the indicator function of a closed convex of H. First, he proves the existence of a global solution (in an adequate weak sense) for some specific data. When \(\partial \phi\) is a positive self- adjoint linear operator in H, he obtains an existence theorem for energy conserving solutions. Restricting to a more specific class of energy conserving solutions, he gets existence and uniqueness. These results extend previous ones obtained by M. Schatzman. Two applications are briefly cited.