Existence of propagation cone for one-dimensional wave integro-differential operator with fractional-exponential memory function

Abstract

We study a linear Volterra integro-differential operator, which is a one-dimensional wave linear partial differential operator perturbed by an integral operator of the Volterra convolution. The kernel of integral operator is the sum of fractional-exponential functions (Rabotnov functions) with positive coefficients. We establish that the support of fundamental solution of the considered integro-differential operator is localized in the propagation cone of the corresponding one-dimensional wave differential operator. The corresponding Volterra integro-differential equation describes the oscillations of one-dimensional viscous-elastic rod, the heat propagation in media with memory (Gurtin — Pipkin equation) and a series of other important applications.