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

TL;DR

This paper investigates the existence of solutions for a nonlocal Penrose-Fife phase field system with inertial effects, introducing a time discretization scheme and deriving error estimates, advancing understanding of such complex systems.

Contribution

It presents a novel time discretization approach and error analysis for a nonlocal Penrose-Fife system with inertial terms, addressing an open problem in the field.

Findings

Established a time discretization scheme for the system

Derived error estimates between continuous and discrete solutions

Provided insights into the existence of solutions for the system

Abstract

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).