Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms

We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form $\mathrm{\partial }u(x,t)\mathrm{/}\mathrm{\partial }t=\mathrm{\Delta }u(x,t-{\tau }_1)+f(u(x,t),u(x,t-{\tau }_2))$, where ${\tau }_1,{\tau }_2$ are positive constants. We extend the monotone iteration method to systems that satisfy typical monotone conditions by thoroughly studying the sign of the Green function associated with a linear functional differential equation. Namely, we show that for small positive $r$ the functional equation $x^{\mathrm{\prime }\mathrm{\prime }}(t)-a{x}^{\mathrm{\prime }}(t+r)-bx(t+r)=f(t)$, where $a≠0,b>0$ has a unique bounded solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is sufficiently small, $f(t)\ge 0$ for $t\in \mathbb{R}$, then the unique bounded solution $x_f(t)\le 0$ for all $t\in \mathbb{R}$. In the framework of the monotone iteration method that is developed based on this result, upper and lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to show that traveling waves exist for these equations when delays are small in both diffusion and reaction terms. The obtained results appear to be new.