Abstract: The self-ignition induced by an imploding arc-shaped shock wave and the subsequent propagation of the combustion waves are investigated by shock tube experiments and quasi-one-dimensional numerical simulations. A carefully designed transitional tube section is employed to smoothly transform the incident planar shock wave to an imploding arc-shaped shock wave. It is found that the non-uniform gradient environment behind the imploding shock consists of three different regions that are respectively produced by the shock wave propagation in the straight section, the transitional section and the wedge section. With the strengthening of the imploding shock wave, a mild chemical reaction zone with absence of shock waves breaks out at a spot behind the arc shock. The upstream front of the reaction zone moves faster than the Chapman-Jouguet (CJ) detonation speed in the very beginning, and the flow of the reaction products exhibits characteristics of a weak detonation wave. Further analysis indicates that the upstream reaction front is essentially a spontaneous reaction wave instead of a hydrodynamic wave and the moving speed of it is consistent with the reciprocal of the local ignition time gradient. The expandence speed of the reaction zone quickly drops below the CJ speed, accompanying with the emergence of shock waves and a shock-flame complex. The shock-flame complex accelerates and transits to a detonation wave, eventually. Under a certain circumstance and because of the unique induction time gradient environment, the spontaneous reaction wave front may overtake the detonation wave to become the new combustion front. Again, when the speed of the spontaneous wave front drops below the CJ detonation speed, a new detonation wave takes over. In this process, the original detonation wave degenerates to a shock wave propagating in the combustion products.