The main equation of this paper is the special case of equation studied by Il’in and Oleinik for single model equation with convex nonlinearity , by considering f(u)=u^m and nonlinear diffusion, for m>0. We are interested in the stability of the degenerate advection-diffusion equation by dealing with the singular term when u+=0. We first transform the original equation into the traveling waves by using the ansatz transformation. The weighted energy estimates of the transformed equation are then established, where the aim of this weighted function is to avoid the singular term when u+=0. At the final stage, the stability of traveling waves is shown based on the weighted energy estimates and appropriate perturbations.