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Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. (English) Zbl 1117.34060
Consider the differential-difference system $$ u_{t}(x,t) = u(x+1,t) - 2u(x,t) + u(x-1,t) + f(u(x,t)) \tag{dds}$$ 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 $$\aligned cU'(\cdot) = U(\cdot + 1) + U(\cdot - 1) - 2 U(\cdot) + f(U(\cdot)) \quad \text{on } \mathbb R,\\ U(-\infty) = 0, \quad U(+\infty) = 1, \quad 0 \leq U \leq 1 \quad \text{on } \mathbb R.\endaligned \tag{twe}$$ Under the assumption $$f \in C^1([0,1]), \quad f(0) = f(1) = 0 < f(s) \quad \text{for all } s \in (0,1) \tag A$$ 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 $\mathbb R.$ (iii) Any solution $(c,U)$ of ({twe}) satisfies $$ \lim_{x \to -\infty} \frac{U''(x)}{U'(x)} = \lambda, \quad \lim_{x \to -\infty} \frac{f(U(x))}{U'(x)} = \cases c &\text{ if}\, \lambda = 0,\\ f'(0)/\lambda &\text{ otherwise}\endcases.$$ $$ \lim_{x \to \infty} \frac{U''(x)}{U'(x)} = \mu, \quad \lim_{x \to \infty} \frac{f(U(x))}{U'(x)} = \cases c &\text{ if } \mu = 0,\\ f'(0)/\mu &\text{ otherwise}\endcases$$ where $\lambda$ is a nonnegative real root of the characteristic equation $$ c\lambda = e^\lambda + e^{-\lambda} - 2 + f'(0)$$ and $\mu$ is the negative real root of the characteristic equation $$ c\mu = e^\mu + e^{-\mu} - 2 + f'(1).$$ In addition, $\lambda$ 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.

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