Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity

In this paper, we study a class of nonlocal dispersion equation with monostable 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, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0 0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function. These rates are optimal and the stability results significantly develop the existing studies for nonlocal dispersion equations.