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In this paper, we study a class of nonlocal dispersion equation withmonostable nonlinearity in $n$-dimensional space u_t - J\ast u +u+d(u(t,x))=\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} \] where the nonlinear functions $d(u)$ and$b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ isthe heat kernel, and the kernel $J(x)$ satisfies ${\hatJ}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$. Afterestablishing the existence for both the planar traveling waves $\phi(x\cdot{\bfe}+ct)$ 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 provethat, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globallystable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau}$ for$\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ areglobally stable in the algebraic form $t^{-n/\alpha}$. The adopted approach isFourier transform and the weighted energy method with a suitably selectedweight function. These rates are optimal and the stability resultssignificantly develop the existing studies for nonlocal dispersion equations.

Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity