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^{-&#957_&#964t} for some constant &#957_&#964=&#957(&#964)&#62 0, where
&#964&#8805 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 &#964 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.