### Electronic Research Archive

2021, Issue 4: 2599-2618. doi: 10.3934/era.2021003

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

• Received: 01 September 2020 Revised: 01 November 2020 Published: 11 January 2021
• Primary: 35K57, 35B35; Secondary: 92D30

• This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in $L^\infty(\mathbb{R})\times L^\infty(\mathbb{R})$ with the exponential convergence rate $e^{-\mu t}$ for some constant $\mu>0$.

Citation: Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay[J]. Electronic Research Archive, 2021, 29(4): 2599-2618. doi: 10.3934/era.2021003

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• This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in $L^\infty(\mathbb{R})\times L^\infty(\mathbb{R})$ with the exponential convergence rate $e^{-\mu t}$ for some constant $\mu>0$.

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