Similar solutions existence and features for piston problem in gas dynamics

An existence of similar solutions of piston problem for gas dynamic equations is shown. A piston velosity may increase or decrease with time. Initially a gas has zero velocity and pressure, its initial density has non-uniform distribution in space. An ad hoc form of ODE describing similar solutions is used in order to prove the existence. A space-time profiles of solutions and conditions of their non-monotony are obtained.