This paper deals with the gasless combustion problem in an infinite one-dimensional medium. The possibility of the existence of steadily propagating reaction waves is stated for system models with nonzero reaction rates everywhere but at the boundaries. Two forms for the chemical reaction rate dependence on the temperature are considered: the classical Arrhenius, and a modified Arrhenius incorporating and ignition temperature. The equations for the steadily propagating waves are studied in the phase-plane of the temperature and its derivative. The analysis first addresses the question of whether steadily propagating waves are admissible. Then bounds for the propagation velocity are sought and found. The results of the closed-form analysis are successfully tested against numerical experiments. Non-steady propagation regimes are also found, and regions in the parameter space associated with different asymptotic dynamical behaviour are identified.