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

Consider the differential-difference system

${u}_{t}\left(x,t\right)=u\left(x+1,t\right)-2u\left(x,t\right)+u\left(x-1,t\right)+f\left(u\left(x,t\right)\right)\phantom{\rule{2.em}{0ex}}\left(\mathrm{dds}\right)$

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

$\begin{array}{c}\hfill c{U}^{\text{'}}\left(·\right)=U\left(·+1\right)+U\left(·-1\right)-2U\left(·\right)+f\left(U\left(·\right)\right)\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}ℝ,\\ \hfill U\left(-\infty \right)=0,\phantom{\rule{1.em}{0ex}}U\left(+\infty \right)=1,\phantom{\rule{1.em}{0ex}}0\le U\le 1\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}ℝ·\end{array}\phantom{\rule{2.em}{0ex}}\left(\mathrm{twe}\right)$

Under the assumption

$f\in {C}^{1}\left(\left[0,1\right]\right),\phantom{\rule{1.em}{0ex}}f\left(0\right)=f\left(1\right)=0

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. ${U}^{\text{'}}>0$ on $ℝ·$

(iii) Any solution $\left(c,U\right)$ of (twe) satisfies

$\underset{x\to -\infty }{lim}\frac{{U}^{\text{'}\text{'}}\left(x\right)}{{U}^{\text{'}}\left(x\right)}=\lambda ,\phantom{\rule{1.em}{0ex}}\underset{x\to -\infty }{lim}\frac{f\left(U\left(x\right)\right)}{{U}^{\text{'}}\left(x\right)}=\left\{\begin{array}{cc}c\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{0.166667em}{0ex}}\lambda =0,\hfill \\ {f}^{\text{'}}\left(0\right)/\lambda \hfill & \phantom{\rule{4.pt}{0ex}}\text{otherwise}\hfill \end{array}\right\·$
$\underset{x\to \infty }{lim}\frac{{U}^{\text{'}\text{'}}\left(x\right)}{{U}^{\text{'}}\left(x\right)}=\mu ,\phantom{\rule{1.em}{0ex}}\underset{x\to \infty }{lim}\frac{f\left(U\left(x\right)\right)}{{U}^{\text{'}}\left(x\right)}=\left\{\begin{array}{cc}c\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}\mu =0,\hfill \\ {f}^{\text{'}}\left(0\right)/\mu \hfill & \phantom{\rule{4.pt}{0ex}}\text{otherwise}\hfill \end{array}\right\$

where $\lambda$ is a nonnegative real root of the characteristic equation

$c\lambda ={e}^{\lambda }+{e}^{-\lambda }-2+{f}^{\text{'}}\left(0\right)$

and $\mu$ is the negative real root of the characteristic equation

$c\mu ={e}^{\mu }+{e}^{-\mu }-2+{f}^{\text{'}}\left(1\right)·$

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.

MSC:
 34K10 Boundary value problems for functional-differential equations 35K57 Reaction-diffusion equations
lattice dynamics