The problem of a piston moving in a gas with energy sources

The process of compressing a gas by a flat piston, assuming that the velocity of the piston is a power function of time, is analyzed. The initial gas density $\rho$ is distributed in the space as $\rho=\rho_1m^l$, where $m$ is the mass Lagrangian variable. The processes of energy release in the medium are modelled by a non-linear source in the energy equation in the form of a power function of temperature and density.