In this talk, I show the recent work to study the propagation of initial disturbance in quadratic Auto-Catalytic chemical reaction in one-dimensional slab geometry, where two chemical species A, called reactant, and B, called auto-catalyst, are involved in the simple scheme A+B->2B. Experiments demonstrate that chemical systems, for which quadratic or cubic autocatalysis forms a key step, can support propagating chemical wave fronts. When the auto-catalyst is introduced locally into an expanse of the reactant which is initially at uniform concentration, the developing reaction is often observed to generate two wave fronts, which propagate outward from the initial reaction zone. We show rigorously that with such an initial setting the spatial region is divided into three regions by the two wave fronts. In the middle expanding region, the reactant is almost consumed so A≈0, whereas in the other two regions there is basically no reaction so B≈0. Most of the chemical reaction takes place near the wave fronts. The detailed characterization of concentrations is given for each of the three zones.