Stability Of Monostable Traveling Wave Solutions Of Reaction-Diffusion Equations With Nonlocal Delay

Traveling wave solutions are a class of stable solutions of reaction-diffusion equations. Many spreading phenomena in nature can be described by traveling wave solutions, such as the finite vibration phenomenon, the spreading of infectious diseases, and so on. As one of the qualitative properties of traveling wave solutions, stability of traveling waves is always the emphasis and difficulty in the traveling waves theory, specially the stability of monostable traveling wave solutions for a system with nonlocal delay and without quasi-monotonicity. The standard theory of scalar equations and the usual methods are no longer applicable due to the coupling of the system; The comparison principle is no longer valid because of the system without quasi-monotonicity. Therefore, there are important theoretical significance and practical value on the investigation of stability of reaction-diffusion(advection) equations with(or without) quasi-monotonicity. Based on the above fact, we address our investigation in the stability of monostable traveling wave solutions for a reaction-advection-diffusion equation with nonlocal delay and a nonlocal system with limited distributed delay and without quasi-monotonicity. The main results in this paper are included as follows.? The stability of monostable traveling fronts of a kind of reaction-advection-diffusion equations with nonlocal delay is established. By the weighted-energy method combining comparison principle, we prove the globally exponential stability of traveling fronts under the large initial perturbation(i.e. the initial perturbation around the traveling wave decays exponentially as x ?-?, but it can be arbitrarily large in other locations) including even the slower waves whose wave speed are close to the critical speed. And the conclusion is extended to a wider kind of reaction-advection-diffusion equations with nonlocal delay.? The stability of monostable traveling waves of a class of reaction-diffusion systems with limited distributed delay and without quasi-monotonicity is established. By the weighted energy method combining continuation method, we prove the monostable traveling waves of the system are exponential stable. Particularly, the initial perturbation is uniformly bounded only at x = +? but may not be vanishing.