Global Stability of Non-monotone Noncritical Traveling Waves for a Discrete Diffusion Equation with a Convolution Type Nonlinearity

Abstract

This paper is concerned with the global stability of non-monotone traveling waves for a discrete diffusion equation with a monostable convolution type nonlinearity. It has been proved by Yang and Zhang (Sci. China Math. 61 (2018), 1789-1806) that all noncritical traveling waves (waves with speeds c > c * , c * is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. In this paper, we further prove that all traveling waves with large speed are globally stable, when the initial perturbations around the waves in a weighted Sobolev space can be arbitrarily large. The approaches adopted are the nonlinear Halanay's inequality, the technical weighted energy method and Fourier's transform.

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.…”

“…The following maximum principle is needed to obtain the crucial boundedness estimate of ( V 1 , V 2 ), which has been proved in [17,Lemma 3.4]. Lemma 3.6.…”

“…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. 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.…”

Propagation dynamics of a discrete diffusive equation with non‐local delay

“…When the kernel function $$$ \beta \left(\cdotp \right) $$$ is symmetric, the stability of traveling wave solutions of Equation () has been studied, see previous works [4–7]. Tian et al [5] and Yang et al [6] proved the local stability of traveling waves of () by the technical weighted energy method.…”

“…If the kernel function $$$ \beta \left(\cdotp \right) $$$ satisfies $$$$ \sum \limits_{k\in \mathrm{\mathbb{Z}}}\beta (k)=1,\sum \limits_{k\in \mathrm{\mathbb{Z}}}\beta (k){e}^{-\lambda k}<\infty \kern0.5em \mathrm{for}\ \mathrm{any}\kern0.5em \lambda >0, $$$$ Yang and Zhang [7] obtained the local stability of all non‐critical traveling waves ( $$$ c>{c}_{\ast } $$$, where $$$ {c}_{\ast } $$$ is the minimal wave speed) of () by using the technical weighted energy method. Furthermore, Su and Zhang [4] established the global stability of all non‐critical traveling waves ( $$$ c>{c}_{\ast } $$$) of () through the anti‐weighted technique.…”