Some decay results in Timoshenko beam with microtemperatures of type III

Abstract The governing equations of the thermoelastic Timoshenko beam with microtemperatures are derived based on type III entropy balance. Our development of this theory is prompted by the increasing use of materials that exhibit thermal variations at the microstructure level. According to this theory, thermal and microthermal waves can propagate with energy dissipation at finite speeds. The existence and uniqueness of weak and classical solutions to the derived model is established by means of semigroup theory. Using the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is not analytic. Then, using the multiplier technique, we show the exponential stability result when (5) holds (when the wave speeds are equal) and the polynomial stability result when (5) does not hold. In the latter case, we prove using a result by Borichev and Tomilov that the solutions decay polynomially at an optimal rate of $t^{-1/2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> .