Planar Traveling Waves For Nonlocal Dispersion Equation With Monostable Nonlinearity

Abstract

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_β(y) b(u(t-τ,x-y)) dy, u(s,x)=u_0(s,x), s\in[-τ,0], \ x\in \mathbb{R}^n} \] 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 ${\hat J}(ξ)=1-\mathcal{K}|ξ|^α+o(|ξ|^α)$ for $0<α\le 2$. After establishing the existence for both the planar traveling waves $ϕ(x\cdot{\bf e}+ct)$ 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 prove that, all noncritical planar wavefronts $ϕ(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/α}e^{-μ_τ}$ for $μ_τ>0$, and the critical wavefronts $ϕ(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/α}$. 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.