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Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. (English) Zbl 1117.34060

Consider the differential-difference system

u t (x,t)=u(x+1,t)-2u(x,t)+u(x-1,t)+f(u(x,t))( dds )

where f(0)=f(1)=0· A travelling wave with speed c for (dds) is a solution of the form u(x,t)=U(x+ct), and the paper is concerned with the ones satisfying

cU ' (·)=U(·+1)+U(·-1)-2U(·)+f(U(·))on,U(-)=0,U(+)=1,0U1on·( twe )

Under the assumption

fC 1 ([0,1]),f(0)=f(1)=0<f(s)foralls(0,1)(A)

the authors prove that

(i) Wave profiles of a given speed are unique up to a translation.

(ii) Any wave profile is monotone, i.e. U ' >0 on ·

(iii) Any solution (c,U) of (twe) satisfies

lim x- U '' (x) U ' (x)=λ,lim x- f(U(x)) U ' (x)=cifλ=0,f ' (0)/λotherwise·
lim x U '' (x) U ' (x)=μ,lim x f(U(x)) U ' (x)=cifμ=0,f ' (0)/μotherwise

where λ is a nonnegative real root of the characteristic equation

cλ=e λ +e -λ -2+f ' (0)

and μ is the negative real root of the characteristic equation

cμ=e μ +e -μ -2+f ' (1)·

In addition, λ is the smallest root when c>c min and the largest root when c=c min · This improves and completes earlier results of X. Chen and J. S. Guo.

MSC:
34K10Boundary value problems for functional-differential equations
35K57Reaction-diffusion equations