Space and time estimates of second gradient thermal problems

We consider the space and time decays of certain problems within the second gradient thermal law. Notably, for this thermal theory, the exponential time decay is precluded. First, the time estimates of polynomial type are obtained for both the thermal equation and the one-dimensional thermoelastic system, where the impossibility of localization with respect to time is also established. Then, the space estimates are deduced for the multidimensional thermoelastic problem, which allow to show the exponential decay of the energy.