In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equationon a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to ahyperbolic partial differential equation with a fourth-order derivative with respect to time.First, we investigate the spatial evolution of solutions of an initial boundary-value problemwith zero boundary conditions on the lateral surface of the cylinder. Under a boundednessrestriction on the initial data, an energy estimate is obtained. An upper bound for theamplitude term in this estimate in terms of the initial and boundary data is also established.For the case of zero initial conditions, a more explicit estimate is obtained whichshows that solutions decay exponentially along certain spatial–time lines. A class ofnon-standard problems is also considered for which the temperature and its first two timederivatives at a fixed time T0 are assumed proportional to their initial values. These resultsare relevant in the context of the Saint–Venant Principle for heat conduction problems.