Time discretization of a nonlocal phase-field system with inertial term

Time discretizations of phase-field systems have been studied. For example, a time discretization and an error estimate for a parabolic-parabolic phase-field system have been studied by Colli--K. [Commun. Pure Appl. Anal. 18 (2019)]. Also, a time discretization and an error estimate for a simultaneous abstract evolution equation applying parabolic-hyperbolic phase field systems and the linearized equations of coupled sound and heat flow have been studied (see K. [ESAIM Math. Model. Numer. Anal.54 (2020), Electron. J. Differential Equations 2020, Paper No. 96]). On the other hand, although existence, continuous dependence estimates and behavior of solutions to nonlocal phase-field systems with inertial terms have been studied by Grasselli--Petzeltov\'a--Schimperna [Quart. Appl. Math. 65 (2007)], time discretizations of these systems seem to be not studied yet. In this paper we focus on employing a time discretization scheme for a nonlocal phase-field system with inertial term and establishing an error estimate for the difference between continuous and discrete solutions