In this paper we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and three-phase-lag theories, reflecting Saint-Venant's principle. In a recent paper, two families of cases for high order partial differential equations were studied. Here we investigate a third family of cases which corresponds to the fact that a certain condition on the time derivative must be satis ed. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén-Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality.