  In this paper we study the time differential dual-phase-lag model of heat
conduction incorporating the microstructural interaction effect in the
fast-transient process of heat transport. We analyse the influence of the delay
times upon some qualitative properties of the solutions of the initial boundary
value problems associated to such a model. Thus, the uniqueness results are
established under the assumption that the conductivity tensor is positive
definite and the delay times $\tau_q$ and $\tau_T$ vary in the set $\{0\leq
\tau_q\leq 2\tau_T\}\cup \{0<2\tau_T< \tau_q\}$. For the continuous dependence
problem we establish two different estimates. The first one is obtained for the
delay times with $0\leq \tau_q \leq 2\tau_T$, which agrees with the
thermodynamic restrictions on the model in concern, and the solutions are
stable. The second estimate is established for the delay times with $0<2\tau_T<
\tau_q$ and it allows the solutions to have an exponential growth in time. The
spatial behavior of the transient solutions and the steady-state vibrations is
also addressed. For the transient solutions we establish a theorem of influence
domain, under the assumption that the delay times are in $\left\{0<\tau_q\leq
2\tau_T\right\}\cup \left\{0<2\tau_T<\tau_q\right\}$. While for the amplitude
of the harmonic vibrations we obtain an exponential decay estimate of
Saint-Venant type, provided the frequency of vibration is lower than a critical
value and without any restrictions upon the delay times.
