On a class of implicit evolution variational inequalities (Q1901103)

Let \(V\subset H\subset V'\) be an evolution triple, where \(V\) and \(H\) are Hilbert spaces. The Cauchy problem for the following nonlinear evolution equation \[ (I+ \partial\phi)(u'(t))+ Lu(t) \ni F(t, u(t)),\quad 0<t< T \] is studied. Here \(\partial\phi\) denotes the subdifferential of a proper, convex, and lower-semicontinuous function \(\phi: V\to (- \infty, +\infty]\), \(L: V\to V'\) is a linear, bounded, and selfadjoint operator, and \(F: (0, T)\times V\to V'\) is a nonlinear function. Qualitative results (via finite differences in time) and applications to PDEs are discussed.