Existence of traveling wave solutions for a reaction–diffusion equation with distributed delays

In this paper, we consider a reaction–diffusion equation with distributed delays. Special attention is paid to the existence of traveling wave solutions. For special delay kernels, using the linear chain trick and the geometric singular perturbation theory, we investigate a natural connection between the existence of traveling wave solutions for the reaction–diffusion equation with distributed delays and the existence of traveling wave solutions for the corresponding undelayed reaction–diffusion equation. We show that if the corresponding undelayed reaction–diffusion equation has a traveling wave solution, then, for any sufficiently small delay, the delayed reaction–diffusion equation also has a traveling wave solution.