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\[{cases} u_t - J u +u+d(u(t,x))= ∫R^n f_β(y) b(u(t-τ,x-y)) dy, \(s,x)=u_0(s,x), \ \ s∈[-τ,0], \ x∈ R^n,{cases}\]where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, f_β(x) is the heat kernel, and the kernel $J(x)$ satisfies J(ξ)=1-K|ξ|^α+o(|ξ|^α) for $00$. After establishing the existence for both the planar traveling waves φ(x· e+ct) for c≥ 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 φ(x· e+ct) are globally stable withthe exponential convergence rate t-n/αe-μ_τ t for μ_τ>0, and the criticalwavefronts φ(x· e+c_*t) are globally stable in the algebraic form t-n/α, 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.