We present an extensive analysis on the spatial behavior of the solutions within the
three-phase-lags model of a rigid heat conductor for a semi-infinite cylinder excited
on its base. The relaxation time of the temperature gradient has a special significance
here, namely (i) in the absence of this relaxation time, we manage to highlight a
theorem of the domain of influence, that is, outside of a region adjacent to the
charged base of the cylinder, all the thermal activity is vanishing, (ii) instead,
when the relaxation time of the temperature gradient is present, then it is
no longer possible to highlight the area of influence, but we can notice the
Saint Venant’s effect. In this latter situation, we are able to describe the
exponential decay with respect to the distance from the charged base for a
measure of the solution, having a suitable time-dependent exponent to show
the rapid decay of the effects when a small time leak has occurred. For the
situations when the base of the cylinder is excited for a longer time, both
the result expressing the domain of influence and the Saint Venant-type
exponential decrease estimate provide insufficient information regarding the spatial
behavior along the generator of cylinder. To deal with this shortcoming, we
establish exponential decay estimates with a time-independent exponent
that can be used to describe the spatial behavior even inside the domain of
influence.
Keywords
three-phase-lag model, rigid conductor, spatial behavior,
domain of influence, Saint Venant's exponential decay
