Decay rates of the solution to the Cauchy problem of the type III Timoshenko model without any mechanical damping

In this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction 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 and, in the case of nonequal wave speeds, are of regularity–loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.