Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

In this paper, we study a class of nonlocal dispersionequation 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 $00$. 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 withthe exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the criticalwavefronts $\phi(x\cdot{\bf e}+c_*t)$ 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 transformand the weighted energymethod with a suitably selected weight function.

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