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Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. (English) Zbl 1010.39004

The authors study the existence and asymptotic stability of traveling waves \[ \dot u_j= [g(u_{j+1})+ q(u_{j-1})- 2g(u_j)]+ f(u_j),\tag{\(*\)} \] where \(j\) is an integer. By a traveling wave to \((*)\) with speed \(c> 0\) they mean a solution to \((*)\) satisfying \(u_j(1/c)= u_{j-1}(0)\) for all \(j\). Of particular interest are the cases \(g(u)= du^p\) with \(d> 0\), \(p\geq 1\) and \(f(u)= u- u^2\).

MSC:

39A12 Discrete version of topics in analysis
39A11 Stability of difference equations (MSC2000)
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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