Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

Abstract

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of $c\ge c^*$ for the degenerate reaction–diffusion equation without delay, where $c^*>0$ is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay $\tau >0$. Furthermore, we prove the global existence and uniqueness of $C^{\alpha ,\beta }$-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted $L^1$-space. The exponential convergence rate is also derived.