MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION

Zhaohai Ma; Xin Wu; Rong Yuan; Yang Wang

doi:10.11948/2156-907X.20180190

2019 Volume 9 Issue 3

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Zhaohai Ma, Xin Wu, Rong Yuan, Yang Wang. MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 962-980. doi: 10.11948/2156-907X.20180190

Citation:

Zhaohai Ma, Xin Wu, Rong Yuan, Yang Wang. MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 962-980. doi: 10.11948/2156-907X.20180190

MULTIDIMENSIONAL STABILITY OF PLANAR WAVES FOR DELAYED REACTION-DIFFUSION EQUATION WITH NONLOCAL DIFFUSION

1.
School of Science, China University of Geosciences, Beijing 100083, China
2.
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
3.
School of Sciences, East China Jiao Tong University, Nanchang 330013, China
4.
School of Mathematical Sciences, Shanxi University, Shanxi 030006, China

Corresponding author: Email address: xinw2015@mail.bnu.edu.cn(X. Wu)
Fund Project: The authors were supported by National Natural Science Foundation of China

Abstract

Abstract

In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$-dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\mathbb{R}^n )$ in the form of ${\rm{e}}^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$($\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\mathbb{R}^n)$ space and $H^k(\mathbb{R}^n) (k \geq [\frac{n+1}{2}])$ space.
- Multidimensional stability /
- planar waves /
- nonlocal diffusion /
- weighted energy /
- Fourier transform
MSC: 35C07, 92D25, 35B35

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References

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