Traveling waves in reaction-diffusion equations with delay in the diffusion term

We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form u (x, t) / t = Δu (x, t-τ₁) +f (u (x, t), u (x, t-τ₂) ), where τ₁, τ₂ 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'' (t) -ax' (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) 0 for t R, then the unique bounded solution xf (t) 0 for all t 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.