Decay for thermoelastic green-lindsay plates in bounded and unbounded domains

We consider equations describing the thermoelastic behavior of plates modeled in the Green-Lindsay sense. This is done with two different type of couplings of the fourth-order plate Kirchhoff-type plate equation to a second-order heat equation of Cattaneo type, once of second, and once of first order. We investigate both systems for bounded domains and for the Cauchy problem, asking for exponential stability in bounded domains resp. polynomial decay rates for the Cauchy problem. It turns out that one system is exponentially stable, while the other is not, and that, in correspondence, one does not have and the other one has regularity loss in the Cauchy problem. This provides a new interesting example where the different couplings lead to qualitatively different behavior, as known before for classical thermoelastic plates, for Timoshenko systems, for porous elasticity or for plates with two temperatures, with Fourier resp. Cattaneo heat conduction. The optimality of the decay rates obtained is also proved.