An analytical model for heating matter in the semi-infinite space by a high-energy ion flow is presented. The obtained solution obtained describes the temporal evolution of the thermodynamics and hydrodynamics of plasma in the heated region. It counts for the general stopping power dependence on plasma temperature. While the matter is heated, the mean free path of heating ions changes functionally with temperature. Namely, it does not depend on the plasma temperature at the beginning and increases with the temperature after it passes a thermal threshold. At this threshold, the thermal velocity of plasma electrons is equal to the velocity of the ions in the heating beam. The solution shows that the most advantageous regime for the initialization of the quasi-stationary shock wave realizes when the ion energy is equal to the thermal threshold. This regime corresponds to the maximum shock pressure and the slowest subsequent pressure decay with time. Special attention is devoted to the powerful shock wave driven by the laser-accelerated fast ion beam. For example, a Gbar shock pressure may be achieved by heating an aluminum target by a proton beam with intensity of about and particle energy of about 4 MeV.