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

N. A. Rautian

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow Center of Fundamental and Applied Mathematics, GSP–1, Leninskie gory 1, 119991, Moscow, Russia

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.

Keywords: Volterra integro–partial differential operator, fundamental solution, Fourier — Laplace transform, fractional–exponential function.

UDC: 517.968.72

MSC: 47G20, 45K05, 35R09

Received: 03.03.2025

 English version: Ufa Mathematical Journal, 2025, 17:4, 81–94