Thermodynamic properties and stability for the heat flux equation with linear memory

Abstract

Within the linearized theory of heat conduction with fading memory, some restrictions on the constitutive equations are found as a direct consequence of thermodynamic principles. Such restrictions allow us to obtain existence, uniqueness, and stability results for the solution to the heat flux equation. Both problems, which respectively occur when the instantaneous conductivity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{k_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is positive or vanishes, are considered.