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 ut − J ∗ u+ u+ d(u(t, x)) = ∫ Rn fβ(y)b(u(t− τ, x− y))dy, u(s, x) = u0(s, x), s ∈ [−τ, 0], x ∈ Rn, where the nonlinear functions d(u) and b(u) possess the monostable characters like Fisher-KPP type, fβ(x) is the heat kernel, and the kernel J(x) satisfies Ĵ(ξ) = 1 − K|ξ|α + o(|ξ|α) for 0 < α ≤ 2 and K > 0. 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 with the exponential convergence rate t−n/αe−μτ t for μτ > 0, and the critical wavefronts 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 FisherKPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.