Existence for a nonlocal Penrose--Fife type phase field system with inertial term

This article deals with a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove existence of solutions in reference to Colli--Grasselli--Ito [Electron. J. Differential Equations 2002, No. 100, 32 pp.] or not (see Remark 1.1). In this paper we introduce a time discretization scheme (see Section 2), pass to the limit as the time step $h$ goes to $0$ and obtain an error estimate for the difference between continuous and discrete solutions (see Section 5).