On the spatial behavior of solutions for three-phase-lag thermal model

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