On the fourth order Cattaneo equation of heat conduction with memory

TL;DR

This paper investigates a less-studied fourth order PDE related to the Cattaneo heat equation, analyzing its well-posedness and asymptotic behavior under parameter limits, especially with nonhomogeneous boundary conditions.

Contribution

It introduces and studies a novel fourth order heat conduction equation with memory, focusing on its mathematical properties and boundary condition effects.

Findings

Established well-posedness of the equation

Analyzed asymptotic behavior as parameters become singular

Highlighted differences from classical second order models

Abstract

The well known heat equation with finite speed of propagation proposed by Cattaneo is obtained by a more general fourth order PDE when a certain (small) parameters is put equal to zero. It seems that this fourth order equation has been essentially overlooked in the literature, in particular when it is subject to non homogeneous boundary conditions. In this paper we examine its well posedness and the asymptotic behavior when certain coefficients tend to singular values.