A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity

TL;DR

This paper establishes the existence, boundedness, and uniqueness of solutions for a complex integrodifferential model describing nonisothermal multi-phase transitions with temperature-dependent properties and nonconstant specific heat and heat conductivity.

Contribution

It introduces a novel mathematical framework for multi-phase systems with temperature-dependent parameters, proving key properties like existence, boundedness, and uniqueness.

Findings

Proved existence and boundedness of solutions.

Established uniqueness and continuous dependence on data.

Handled nonconstant specific heat and heat conductivity.

Abstract

In this paper, we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi-phase transitions under nonhomogeneous third type boundary conditions. The system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro-differential inclusion governing the (vectorial) phase-parameter dynamics. The specific heat and the heat conductivity are allowed to depend both on the order parameter and on the absolute temperature of the system, and the convex component of the free energy may or may not be singular. Uniqueness and continuous data dependence are also proved under additional assumptions.