Global existence and large-time behavior of smooth solutions to the Timoshenko-Fourier system in thermoelasticity

In earlier works (see for example Kawashima (1984) and Umeda et al. (1984)), S. Kawashima and his collaborators gave a complete theory on the global existence and large-time behavior of smooth solutions to a rather general class of symmetric hyperbolic-paraboliccomposite systems, however, so far there are no general results in the case of non-symmetric dissipation. This paper is devoted to investigate the Timoshenko-Fourier system in thermoelasticity, which describes the dynamics of elastic wave propagation and thermal dissipation in a vibrating beam. Indeed, the Timoshenko-Fourier system admits a non-symmetric dissipation, furthermore, it is observed that the dissipative mechanism of regularity-loss type will occur if wave speeds are different, which makes those classical energy methods in Ide and Kawashima (2008) and Racke and Said-Houari (2012) cannot work. Consequently, we perform a new technique (say, Lp-Lq-Lr estimates) to get the optimal decay rate of L1-L2 type for global smooth solutions.