This note is devoted to the study of the time decay of the onedimensional dual-phase-lag thermoelasticity. In this theory two delay parameters τand τare proposed. It is known that the system is exponentially stable if τ< 2τ. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τ= 2τas 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 τis spatially dependent.