This note is devoted to the study of the time decay of the onedimensional dual-phase-lag thermoelasticity. In this theory two delay parameters tq and t¿ are proposed. It is known that the system is exponentially stable if tq < 2t¿ . We here make two new contributions to this problem. First, we prove the polynomial stability in the case that tq = 2t¿ as well the optimality of this decay rate. Second, we prove that the exponential stability remains true even if the inequality only holds in a proper sub-interval of the spatial domain, when t¿ is spatially dependent.
In this paper we investigate the spatial behavior of the solutions for
a theory for the heat conduction with one delay term. We obtain a Phragm én-
Lindelöf type alternative. That is, the solutions either decay in an exponential
way or blow-up at in nity in an exponential way. We also show how to obtain
an upper bound for the amplitude term. Later we point out how to extend
the results to a thermoelastic problem. We nish the paper by considering
the equation obtained by the Taylor approximation to the delay term. A
Phragm én-Lindelöf type alternative is obtained for the forward and backward
in time equations.
We consider a model for a damped spring-mass system that is a strongly damped wave equation with dynamic boundary conditions. In a previous paper we showed that for some values of the parameters of the model, the large time behaviour of the solutions is the same as for a classical spring-mass damper ODE. Here we use spectral analysis to show that for other values of the parameters, still of physical relevance and related to the effect of the spring inner viscosity, the limit behaviours are very different from that classical ODE.