Existence for a singular nonlocal phase field system with inertial term

TL;DR

This paper proves the existence of solutions for a complex singular nonlocal phase field system with inertial effects, involving challenging mathematical features like logarithmic temperature terms and nonlocal diffusion.

Contribution

It introduces a novel approach to establish solution existence for a difficult nonlocal phase field system with inertial and singular terms.

Findings

Existence of solutions established for the system.

Key estimate enabling mathematical analysis.

Handles singular logarithmic temperature term.

Abstract

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 $\varphi$ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.1).