On the decay rate of a Timoshenko system with dual‐phase‐lag thermoelasticity in unbounded domains

In this paper, we investigate the Cauchy problem for a dual‐phase‐lag (DPL) thermoelastic Timoshenko system with a DPL heat conduction. This DPL model, which includes two thermal relaxation times, τq$\tau _{q}$ and τθ$\tau _{\theta }$, describes non‐instantaneous heat propagation. We first study the decay properties of the system using the energy method in Fourier space, by constructing an appropriate Lyapunov functional. Then, we prove that the L2$L^{2}$‐norm of the solution decays with the rate (1+t)−1/8$(1+t)^{-1/8}$, with some regularity loss and under the assumption 2τθ>τq$2\tau _{\theta }>\tau _{q}$.