A numerical model to explore the possibility that the dissipation of two upward propagating internal gravity waves, identified in the temperature measurements of the Galileo Probe, provide the energy to maintain Jupiter's high thermospheric temperatures βΌ900 K is used. The propagation and dissipation of the gravity waves are simulated by a full-wave model that is used to calculate the thermal mean-state forcing. The observed temperature is the result of this forcing and other energy sources. The equation of heat transfer, including the effects of eddy and molecular thermal diffusion, is solved to provide the gravity wave contribution to the steady-state temperature distribution. For the smallest values of eddy diffusion considered, the waves can heat the entire thermosphere, while for the largest values of eddy diffusion, the waves can cool the entire thermosphere. However, in all cases considered the net heating and cooling effects are not large, being typically βΌ15-20 K or less. To the extent that the Galileo data characterize gravity waves in Jupiter's atmosphere, gravity wave dissipation is unlikely to be the source of energy maintaining Jupiter's high thermospheric temperatures. Other waves not identified in the Galileo data or other energy sources must be responsible for heating the jovian thermosphere. We demonstrate that the viscous heating can only be calculated using the viscous stress tensor, and that the use of the wave mechanical energy flux divergence for this purpose in previous studies is invalid.