Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with microtemperature effect

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with temperature and microtemperature effect. The heat conduction is given by the theory of Lord–Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord–Shulman’s theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow, we prove our main result under suitable assumption.