Global stability of traveling waves for Nagumo equations with degenerate diffusion

This paper is concerned with the global nonlinear stability with possibly large perturbations of the unique sharp / smooth traveling waves for the degenerate diffusion equations with Nagumo (bistable) reaction. Two technical issues arise in this study. One is the shortage of weak regularity of sharp traveling waves, the other difficulty is the non-absorbing initial-perturbation around the smooth traveling waves at the far field $x=+\infty$. For the sharp traveling wave case, we technically construct weak sub- and super-solutions with semi-compact supports via translation and scaling of the unique sharp traveling wave to characterize the motion of the steep moving edges and avoid the weak regularity of the solution near the steep edges. For the smooth traveling wave case, we artfully combine both the translation and scaling type sub- and super-solutions and the translation and superposition type sub- and super-solutions in a systematical manner.