Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay
The paper is concerned with the study of the traveling wave fronts of the discrete quasilinear equation with delay \[ u_t(x,t)=D_2[g(u)](x,t)+f(u(x,t),u(x,t-\tau)),\quad \forall\,x\in\mathbb{R},\quad t>0,\tag{1} \] where \(\tau\geq 0\) and \(D_2\) is the linear operator from \(C(\mathbb{R})\) to \(C(\mathbb{R})\) defined by \[ D_2[\phi](x,t)=\phi(x+1,t)+\phi(x-1,t)-2\phi(x,t),\quad \forall \,x\in\mathbb{R}. \] The traveling wave problem consists in finding \((c,U)\in\mathbb{R}\times C^1(\mathbb{R})\) satisfying the equation \[ c U'-D_2[g(U)]-f(U,U(\cdot-c\tau))=0,\quad U'>0,\quad \text{on}\quad \mathbb{R}\tag{2} \] and the boundary conditions \[ U(-\infty)=0,\quad U(\infty)=1.\tag{3} \] By using the method of super-solutions and sub-solutions and the monotone iteration technique, the authors establish the existence of the traveling wave fronts for problem (1) (solutions of (2)-(3)), and then they show that the traveling wave front is unique up to a translation. In the case \(g(u)=du\) the authors use the comparison principle and the squeezing technique to prove that the traveling wave front of (1) is globally asymptotically stable with phase shift.
scientific article; zbMATH DE number 5692919