Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays
In this paper, the authors consider the stability of traveling wave solutions for the following class of nonlinear cellular neural networks with distributed time delays: \begin{align*} x_n'(t)=-x_n(t)&+\sum_{i=1}^m \alpha_i\int_0^{\tau}J_i(y)f(x_{n-i}(t-y))\,dy+a \int_0^{\tau}J_{m+1}(y)f(x_{n}(t-y))\,dy\\ &+\sum_{j=1}^{\ell} \beta_j\int_0^{\tau}J_{m+1+j}(y)f(x_{n+j}(t-y))\,dy, \forall n\in \mathbb{Z}, \: m,\ell\in \mathbb{N}, \end{align*} where \(x_n\) represents the \(n\)-th cell of the neural, \(f\in C([0, +\infty), [0, +\infty))\) denotes the output function of the cells, the nonnegative constants \(a\), \(\alpha_i\) \((i =1,\ldots,m)\) and \(\beta_j\) (\(j = 1,\ldots,\ell)\) are related to the synaptic weights of self-feedback and its neighbors to the cells, \(\tau>0\) is a constant delay, and the functions \(J_i:[0,\tau]\to[0,+\infty)\) are piecewise continuous density functions describing the distributed delay effects from the cells. A travelling solution for this infinite system is a solution such that there exist \(c > 0\) (called wave speed) and a function \(\phi: \mathbb{R} \to \mathbb{R}\) smooth enough satisfying that \(x_n(t)=\phi(n-ct),\) for every \(t\in\mathbb{R}\) and \(n\in\mathbb{Z}\). The work is based on the existence of nondecreasing travelling wave solutions satisfying the conditions \( \lim_{\xi\to-\infty} \phi(\xi)=0,\) and \( \lim_{\xi\to+\infty} \phi(\xi)=K.\) To study the stability, the authors prove two kinds of comparison results for the Cauchy problem, and the global existence of solution for two Cauchy problems. Under the appropriate hypotheses, and using some decay estimates, they show that the solutions to the Cauchy problems converge exponentially to the traveling wave solutions when the initial perturbations around the traveling wave solutions belong to suitable weighted spaces.
scientific article; zbMATH DE number 6958717