Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:ut(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),where x∈R, t>0, D,d>0, r⩾0, b∈C1(R) and b(0)=dK-b(K)=0 for some K>0 under monostable assumption. We show that there exists a minimal wave speed c*>0, such that for each c>c* the equation has exactly one travelling wavefront U(x+ct) (up to a translation) satisfying U(-∞)=0,U(+∞)=K and limsupξ→-∞U(ξ)e-Λ1(c)ξ<+∞, where λ=Λ1(c) is the smallest solution to the equation cλ-D[eλ+e-λ-2]+d-b′(0)e-λcr=0. Moreover, the travelling wavefront is strictly monotone and asymptotically stable with phase shift in the sense that if an initial data ϕ∈C(R×[-r,0],[0,K]) satisfies liminfx→+∞ϕ(x,0)>0 and limx→-∞maxs∈[-r,0]|ϕ(x,s)e-Λ1(c)x-ρ0eΛ1(c)cs|=0 for some ρ0∈(0,+∞), then the solution u(x,t) of the corresponding initial value problem satisfies limt→+∞supR|u(·,t)/U(·+ct+ξ0)-1|=0 for some ξ0=ξ0(U,ϕ)∈R.