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

Abstract We study the existence and asymptotic stability of traveling waves to u t =[g(u +1 )+g(u −1 )−2g(u)]+f(u) on R ×(0,∞) , where u = u ( x,t ), u ±1 = u ( x ±1, t ), g = du p ( d >0, p ⩾1) and f = u − u 2 . We show that there exists c >0 such that for each wave speed c > c , there is a traveling wave U ∈ C 1 ( R ), i.e., a solution of the form u = U ( x − ct ). The traveling wave has the property that U (−∞)=1, U ′ R , and lim ξ →∞ U ( ξ ) e λξ =1, where λ =Λ 1 ( c ) is the smallest solution to cλ = f ′(0)+ g ′(0)[ e λ + 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 ( R →[0,1]) satisfies lim inf x →−∞ u ( x ,0)>0 and lim x →∞ u ( x ,0) e λx =1 for some λ ∈(0,Λ 1 ( c )), then lim t →∞ sup R ∣ u (·+ ct , t )/ U (·)−1∣=0 where ( c , U ) is the traveling wave with speed c = C ( λ )={ f ′(0)+ g ′(0)[ e λ + e − λ −2]}/ λ , the inverse of λ =Λ 1 ( c ).