Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

Rui Huang; Ming Mei; Yong Wang

doi:10.3934/dcds.2012.32.3621

Article Contents

2012, Volume 32, Issue 10: 3621-3649. Doi: 10.3934/dcds.2012.32.3621

This issue Previous Article Transport, flux and growth of homoclinic Floer homology Next Article Cone-fields without constant orbit core dimension

Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

1.
School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631
2.
Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2
3.
Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190

Received: May 2011

Revised: December 2011

Published: October 2012

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Abstract

In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space \[ \begin{cases} u_t - J\ast u +u+d(u(t,x))= \displaystyle \int_{\mathbb{R}^n} f_\beta (y) b(u(t-\tau,x-y)) dy, \\ u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n, \end{cases} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$ and $\mathcal{K}>0$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical Fisher-KPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.

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Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D25.

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