Taiwanese Journal of Mathematics

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

Tao Su and Guo-Bao Zhang

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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 $c \gt c_*$, $c_*$ 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.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 21 pages.

Dates
First available in Project Euclid: 9 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1568016022

Digital Object Identifier
doi:10.11650/tjm/190901

Subjects
Primary: 35K57: Reaction-diffusion equations 35C07: Traveling wave solutions 92D25: Population dynamics (general)

Keywords
discrete diffusion equations traveling waves global stability weighted energy method

Citation

Su, Tao; Zhang, Guo-Bao. Global Stability of Non-monotone Noncritical Traveling Waves for a Discrete Diffusion Equation with a Convolution Type Nonlinearity. Taiwanese J. Math., advance publication, 9 September 2019. doi:10.11650/tjm/190901. https://projecteuclid.org/euclid.twjm/1568016022


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