Existence for a singular nonlocal phase field system with inertial term

In this paper we deal with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $\theta$ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln \theta {)}_t$, the inertial term and the nonlocal diffusion term for the order parameter $\phi$ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.1).