Coleman-Gurtin type equations with dynamic boundary conditions

TL;DR

This paper generalizes classical heat conduction theory to include Coleman-Gurtin equations with dynamic boundary conditions, addressing well-posedness with nonlinear and memory effects on both interior and boundary.

Contribution

It introduces a new formulation that encompasses dynamic boundary conditions with memory, extending existing models to more complex and realistic heat conduction scenarios.

Findings

Established well-posedness of the generalized system

Included nonlinear terms with dissipation or balance conditions

Allowed different memory kernels for interior and boundary

Abstract

We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory which includes the usual heat equation subject to a dynamic boundary condition as a special case. We investigate the well-posedness of systems which consist of Coleman-Gurtin type equations subject to dynamic boundary conditions, also with memory. Nonlinear terms are defined on the interior of the domain and on the boundary and subject to either classical dissipation assumptions, or to a nonlinear balance condition in the sense of [11]. Additionally, we do not assume that the interior and the boundary share the same memory kernel.