Evolution Inclusion on a Real Hilbert Space with Quasi-Variational Structure for Inner Products

We consider a Cauchy problem of an abstract evolution inclusion on a real Hilbert space associated with subdifferentials of time-dependent proper lower semicontinuous convex functions. The abstract evolution inclusion, which is treated in this paper, contains not only convex functions but also inner products of the Hilbert space depending on unknown functions. Especially, we call such structures for convex functions and inner products "quasi-variational structures for convex functions and inner products". The main purposes of this paper are to show the existence of global-in-time solutions to the Cauchy problem, which has a quasi-variational structure, and to apply this existence result to a phase field model with a quasi-variational boundary condition.