Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay

Global stability of traveling waves for a spatially discrete diffusion system with time delay

“…Thus, the classical methods, such as the monotone technique and the Fourier transform cannot be applied directly to establish the decay estimate for ( V 1 , V 2 ). Motivated by [15,28,17,23], we introduce a new method which can be described as follows.…”

“…This method is based on some key observations for the structure of the govern equations and the anti-weighted energy method together with the Fourier transform. Later on, Zhang [28] and Xu et al [23], respectively, applied this method successfully to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed.…”

“…More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed. Motivated by the works [15,28,23,17,18], in this paper, we shall extend this method to study the global stability of traveling waves of spatial discrete diffusion system (1) without quasi-monotonicity.…”

Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity

“…The stability of traveling waves for various evolution equations has been extensively studied. We refer the readers to [4,5,9,10,11,13,14,15,18,19,21,26] for reaction-diffusion equations and to [8,12,17,27,28,29,30] for nonlocal dispersal equations. Note that when the evolution equations are non-monotone, the comparison principle is not applicable.…”

“…Recently, Mei et al [16] established the global stability for the oscillatory traveling waves of local Nicholson's blowflies equations by using the anti-weighted energy method together with the Fourier transform. Zhang [29] applied this method to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. Motivated by [12,16,29], we shall extend this method to the study of global stability of traveling waves of reaction-diffusion system (1.1) without quasi-monotonicity.…”