The non-stationary heat conduction in an infinitely long solid cylinder with a time dependent boundary heat Aux is studied for a material with a non-vanishing thermal relaxation time. An analytical solution of the hyperbolic energy equation together with its boundary and initial conditions is obtained by the Laplace transform method. The temperature distribution and the heat flux density distribution are studied both for a constant boundary heat flux and for an exponentially decaying boundary heat flux. The compatibility of these distributions with the local equilibrium hypothesis is analysed. (C) 1997 Elsevier Science Ltd.