Global front solutions converging to an asymptotic traveling wave in a time-fractional Amari model

Abstract

We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing $C^1$ front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.