Monotone plane wave of the reaction-diffusion systems with delays appearing in both diffusion and reaction terms

In this paper, we consider a class of reaction-diffusion systems with delays appearing in both diffusion and reaction terms. By analyzing the properties of a key operator derived from the Perron's theorem, and employing Schauder's fixed point theorem, some existence results of the monotone plane waves for these systems are established. The method we adopt to prove the compactness of the key operators is different from that in the existing literature which discusses the existence of traveling wave solutions for such equations. Moreover, we provides a detailed derivation of the expression of bounded solutions for the second-order functional differential equation $ X^{\prime \prime}(t)-p X^{\prime}(t+r)-q X(t+r) = S(t) $, which plays an important role in the study of traveling wave solutions for reaction-diffusion equations with diffusion delay. To illustrate the feasibility of the theoretical results, we investigate the existence of monotone plane waves for the reaction-diffusion system with Belousov-Zhabotinskii reaction, diffusion delay, and reaction delay.