Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Faye, Grégory

doi:10.3934/dcds.2016.36.2473

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…Furthermore, they also found a special solution that oscillates permanently between two planar traveling waves, which implicits that planar waves are not asymptotically stable under more general perturbations. We can refer to [1,5,9,15,27,28,33,36,39,44] for more details about the multidimensional stability of traveling waves.…”

Section: Introductionmentioning

confidence: 99%

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Xue

Liu

2023

Qual. Theory Dyn. Syst.

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In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term in n-dimensional space. It is proved that, all planar traveling waves with speed c > c * are exponentially stable in the form of t − n 2 e −ϵτ σt , where constant σ > 0 and ϵ τ = ϵ(τ ) ∈ (0, 1) is a decreasing function for the time delay τ > 0. While, for the planar traveling waves with speed c = c * , we prove that they are algebraically stable in the form of t − n 2 . The Fourier transform plays a crucial role in transforming the competition systems to a linear delayed differential system. We establish the comparison principle and some estimates in weighted spaces L 1 w (R n ) and W 2,1 w (R n ) to obtain the main results.

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Section: Introductionmentioning

confidence: 99%

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Xue

Liu

2023

Qual. Theory Dyn. Syst.

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Section: Introductionmentioning

confidence: 99%

Stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term

Xue

Liu

2022

Preprint

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In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term in n–dimensional space. It is proved that, all planar traveling waves with speed c > c* are exponentially stable in the form of t−n/2e−ϵτσt, where constant σ > 0 and ϵτ = ϵ(τ ) ∈ (0, 1) is a decreasing function for the time delay τ > 0. While, for the planar traveling waves with speed c = c*, we prove that they are algebraically stable in the form of t−n/2. The Fourier transform plays a crucial role in transforming the competition systems to a linear delayed differential system. We establish the comparison principle and some estimates in weighted spaces L1w(Rn) and Ww2,1(Rn) to obtain the main results.

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“…Very recently, Faye [6] extended the local diffusion equation (1.1) to the following nonlocal diffusion equation and investigated the multidimensional stability of planar traveling waves by using semigroup estimates,…”

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confidence: 99%

Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

Ma¹,

Yuan²,

Wang³

et al. 2019

Communications on Pure &Amp; Applied Analysis

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In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in n-dimensional space. More precisely, we prove that all planar traveling waves with speed c > c * are exponentially stable in L ∞ (R n) in the form of t − n 2α e −ετ σt for some constants σ > 0 and ετ ∈ (0, 1), where ετ = ε(τ) is a decreasing function refer to the time delay τ > 0. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed c = c * , we show that they are algebraically stable in the form of t − n 2α. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

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Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Cited by 7 publications

References 17 publications

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term

Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

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