Existence and Asymptotic Stability of Traveling Waves of Discrete Quasilinear Monostable Equations

We study the existence and asymptotic stability of traveling waves to , where u=u(x,t), u±1=u(x±1,t), g=dup (d>0, p 1) and f=u−u2. We show that there exists c>0 such that for each wave speed c>c, there is a traveling wave U C1( ), i.e., a solution of the form u=U(x−ct). The traveling wave has the property that U(−∞)=1, U′<0 on , and limξ→∞U(ξ)eλξ=1, where λ=Λ1(c) is the smallest solution to cλ=f′(0)+g′(0)[eλ+e−λ−2]. We also show that the traveling wave is globally asymptotically stable in the sense that if an initial value u(•,0) C( →[0,1]) satisfies lim infx→−∞u(x,0)>0 and limx→∞u(x,0)eλx=1 for some λ (0,Λ1(c)), then limt→∞ sup u(•+ct,t)/U(•)−1 =0 where (c,U) is the traveling wave with speed c=C(λ)={f′(0)+g′(0)[eλ+e−λ−2]}/λ, the inverse of λ=Λ1(c).