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 uₜ - J u +u+d (u (t, x) ) = ₑ䂞 f_ (y) b (u (t-, x-y) ) dy, \ (s, x) =u₀ (s, x), \ \ s[-, 0, \ x Rⁿ, cases\]where the nonlinear functions d (u) and b (u) possess the monostable characters like Fisher-KPP type, f_ (x) is the heat kernel, and the kernel J (x) satisfies J () =1-K||^+o (||^) for 00. After establishing the existence for both the planar traveling waves (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 (x e+ct) are globally stable withthe exponential convergence rate t^-n/e^-_ t for _>0, and the criticalwavefronts (x e+c_*t) are globally stable in the algebraic form t^-n/, 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.