Green and Naghdi proposed three theories of thermoelasticity labelled as type I, II and III, respectively. Here we investigate the spatial behaviour of solutions of a problem determined by the type III thermoelasticity. We study the growth/decay alternative in the spatial variable for the linear theory in the case that the thermoelastic deformations and the instant T are constrained to be proportional to the ones at the instant zero. In the case that the solutions decay, an upper bound for the amplitude term by means of the data of the problem can be obtained. We sketch how to obtain it. we finish by noting (through examples) that for several families of cases the solutions can be periodic in the spatial variable and the Phragmen-Lindelof alternative is not satisfied.

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