This paper investigates the heat propagation process in a gas from concentrated energy sources with deposition times, $t'_d$, of the order of the characteristic acoustic time, $t'_a$, across the region where the temperature will be increased by a factor of order unity. Heat propagation takes place by two different mechanisms that act separately in two different neatly defined spatial regions of comparable size. Around the source, we find a conductive region of very high temperature where the spatial pressure variations are negligible. The edge of the resulting strongly heated low-density region appears as a contact surface that acts as a piston for the outer flow, where the pressure disturbances, of order of the ambient pressure in the distinguished regime $t'_d \,{\sim}\, t'_a$ considered here, generate a shock wave that heats up the outer gas as it propagates outwards. The mass and energy balances for the conductive region provide a differential equation linking its pressure with the velocity of its bounding contact surface, which is used, together with the jump conditions across the shock, when integrating the Euler equations for the outer compressible flow. Solutions for the heating history are obtained for point, line and planar sources for different values of the ratio $t'_d/t'_a$, including weak sources with $t'_d \,{\gg}\, t'_a$ and very intense sources with $t'_d \,{\ll}\, t'_a$. The solution determines in particular the temperature profile emerging as the pressure perturbations become negligible for times much larger than the acoustic time.