Evans functions and asymptotic stability of traveling wave solutions (Q5943621)

The author studies the stability of traveling wave solutions to integro-differential equations of the form \[ \partial u_i/\partial t=f_i(u)+\alpha_i\int_{-\infty}^{\infty}K_i(x-y)H(u_i(y,t)-\theta_i) dy, \] \(i=1,2,\dots,n,\) for \(x\in(-\infty,\infty)\) and \(t>0\). Here, \(H\) denotes Heaviside's function and the kernels \(K_i\) are nonnegative, even functions decaying exponentially and satisfying the normalization \(\int_{-\infty}^{\infty}K_i(x) dx=1\). This type of equations appears in modeling of neuronal networks and may consist of billions of equations. In the paper, it is assumed that this system has a family of traveling wave solutions. The main goal is to prove their asymptotic stability using the Evans function approach (cf. e.g. \textit{J. W. Evans} [Indiana Univ. Math. J. 24, 1169-1190 (1975; Zbl 0317.92006)] and the references given there).