2023/01/27 by Barker, William, Van Minh, Nguyen · 1 citation
#35C07 #35K57 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.11504
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\not=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∈ \mathbb R, then the unique bounded solution xf(t)≤ 0 for all t∈ \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.