Regularity of Solutions for the Nonlocal Wave Equation on Periodic Distributions

TL;DR

This paper investigates the regularity properties of solutions to a nonlocal wave equation on periodic distributions, using Fourier multipliers, and explores their convergence to classical wave solutions under certain limits.

Contribution

It introduces a unified Fourier multiplier approach to analyze regularity of solutions for nonlocal wave equations with various kernels, extending to nonlocal pseudo-differential operators.

Findings

Solutions exhibit specific regularity depending on initial data and forcing terms.

Solutions converge to classical wave solutions as nonlocality diminishes or kernel singularity reaches a critical point.

The approach applies to any spatial dimension and includes singular and integrable kernels.

Abstract

This work addresses the regularity of solutions for a nonlocal wave equation over the space of periodic distributions. The spatial operator for the nonlocal wave equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal wave equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal-pseudo differential operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the forcing term. Moreover, solutions of the nonlocal wave equation are shown to converge to the solution of the classical wave equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension.