Traveling Wavefronts in Lattice Differential Equations with Time Delay and Global Interaction

Abstract

In this paper, we study the existence of traveling wave solutions in lattice differential equations with time delay and global interaction

$$\begin{array}{rcl}{ u^{\prime}}_{n}(t)& =& D\sum\limits_{i\in {\mathit{Z}}^{q}\setminus \{0\}}J(i)[{u}_{n-i}(t) - {u}_{n}(t)] \\ & & +F\left ({u}_{n}(t),\sum\limits_{i\in {\mathit{Z}}^{q}}K(i){\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g({u}_{ n-i}(t + \theta ))\right )\end{array}$$

Following an idea in [10], we are able to relate the existence of traveling wavefront to the existence of heteroclinic connecting orbits of the corresponding ordinary delay differential equations

$$u^{\prime}(t) = F\left (u(t),{\int \nolimits \nolimits }_{-r}^{0}\mathrm{d}\eta (\theta )g(u(t + \theta ))\right ).$$

Mathematics Subject Classification 2010(2010): Primary 34K30, 35B40; Secondary 35R10, 58D25