Volume 2, Issue 4
Remark on Stability of Traveling Waves for Nonlocal Fisher-Kpp Equations

MING MEI AND YONG WANG

Int. J. Numer. Anal. Mod. B, 2 (2011), pp. 379-401

Published online: 2011-02

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  • Abstract
This paper is concerned with a class of nonlocal Fisher-KPP type reaction-diffusion equations in n-dimensional space with time-delay. It is proved that, all noncritical planar wavefronts are exponentially stable in the form of t^{-\frac{n}{2}}e^{-ν_τt} for some constant ν_τ=ν(τ)> 0, where τ≥ 0 is the time-delay, while the critical planar wavefronts are algebraically stable in the form of t^{-\frac{n}{2}}. These convergent rates are optimal in the sense with L^1-initial perturbation. The adopted approach is the weighted energy method combining Fourier transform. It is also realized that, the effect of time-delay essentially causes the decay rate of the solution slowly down. These results significantly generalize and develop the existing study [37] for 1-D time-delayed Fisher-KPP type reaction-diffusion equations. When the time-delay τ vanishes, we automatically obtain the exponential stability for the noncritical planar traveling waves and the algebraic stability for the critical planar traveling waves to the regular Fisher-KPP equations.
  • AMS Subject Headings

35K57 34K20 92D25

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@Article{IJNAMB-2-379, author = {MING MEI AND YONG WANG}, title = {Remark on Stability of Traveling Waves for Nonlocal Fisher-Kpp Equations}, journal = {International Journal of Numerical Analysis Modeling Series B}, year = {2011}, volume = {2}, number = {4}, pages = {379--401}, abstract = {This paper is concerned with a class of nonlocal Fisher-KPP type reaction-diffusion equations in n-dimensional space with time-delay. It is proved that, all noncritical planar wavefronts are exponentially stable in the form of t^{-\frac{n}{2}}e^{-ν_τt} for some constant ν_τ=ν(τ)> 0, where τ≥ 0 is the time-delay, while the critical planar wavefronts are algebraically stable in the form of t^{-\frac{n}{2}}. These convergent rates are optimal in the sense with L^1-initial perturbation. The adopted approach is the weighted energy method combining Fourier transform. It is also realized that, the effect of time-delay essentially causes the decay rate of the solution slowly down. These results significantly generalize and develop the existing study [37] for 1-D time-delayed Fisher-KPP type reaction-diffusion equations. When the time-delay τ vanishes, we automatically obtain the exponential stability for the noncritical planar traveling waves and the algebraic stability for the critical planar traveling waves to the regular Fisher-KPP equations.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnamb/319.html} }
TY - JOUR T1 - Remark on Stability of Traveling Waves for Nonlocal Fisher-Kpp Equations AU - MING MEI AND YONG WANG JO - International Journal of Numerical Analysis Modeling Series B VL - 4 SP - 379 EP - 401 PY - 2011 DA - 2011/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnamb/319.html KW - Nonlocal reaction-diffusion equations KW - time delays KW - traveling waves KW - global stability KW - the Fisher-KPP equation KW - L^1-weighted energy KW - Green functions AB - This paper is concerned with a class of nonlocal Fisher-KPP type reaction-diffusion equations in n-dimensional space with time-delay. It is proved that, all noncritical planar wavefronts are exponentially stable in the form of t^{-\frac{n}{2}}e^{-ν_τt} for some constant ν_τ=ν(τ)> 0, where τ≥ 0 is the time-delay, while the critical planar wavefronts are algebraically stable in the form of t^{-\frac{n}{2}}. These convergent rates are optimal in the sense with L^1-initial perturbation. The adopted approach is the weighted energy method combining Fourier transform. It is also realized that, the effect of time-delay essentially causes the decay rate of the solution slowly down. These results significantly generalize and develop the existing study [37] for 1-D time-delayed Fisher-KPP type reaction-diffusion equations. When the time-delay τ vanishes, we automatically obtain the exponential stability for the noncritical planar traveling waves and the algebraic stability for the critical planar traveling waves to the regular Fisher-KPP equations.
MING MEI AND YONG WANG. (2011). Remark on Stability of Traveling Waves for Nonlocal Fisher-Kpp Equations. International Journal of Numerical Analysis Modeling Series B. 2 (4). 379-401. doi:
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