# Global existence for a singular phase field system related to a sliding mode control problem

## TL;DR

This paper proves the existence of solutions for a highly nonlinear singular phase field system with nonlocal and logarithmic nonlinearities, relevant to sliding mode control, using a backward finite differences scheme.

## Contribution

It introduces a novel approach to establish existence results for a complex nonlinear phase field system with nonlocal and logarithmic terms.

## Key findings

- Existence of solutions for the singular phase field system.
- Development of a backward finite differences scheme for analysis.
- Uniform estimates enabling passage to the limit.

## Abstract

In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear; the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step.