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 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.

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