Coleman–Gurtin type equations with dynamic boundary conditions

• Published in: Physica. D Vol. 292-293; pp. 29 - 45
• Main Authors: Gal, Ciprian G., Shomberg, Joseph L.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2015
• Subjects: Balancing; Boundaries; Boundary conditions; Coleman–Gurtin equation; Dissipation; Dynamic boundary conditions with memory; Dynamic tests; Dynamical systems; Heat conduction; Heat equations; Mathematical analysis; Nonlinear dynamics; Nonlinearity; Heat conduction; Coleman–Gurtin equation; Dynamic boundary conditions with memory; Heat equations
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2014.10.008

We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory. 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 Gal (2012). Additionally, we do not assume that the interior and the boundary share the same memory kernel. •We consider boundary conditions with temperature dependent sources/sinks and memory.•We consider memory functions on the boundary and in the interior that differ.•We consider nonlinear terms satisfying nonlinear balance conditions.•We develop a general framework allowing for both weak and smooth initial data.•We extend a Galerkin approximation scheme.