The regularity properties of nonlocal abstract wave equations

TL;DR

This paper investigates the regularity, existence, and uniqueness of solutions for a broad class of nonlocal wave equations involving convolution operators with general kernels, applicable to various physical models.

Contribution

It introduces a general framework for analyzing nonlocal wave equations with operator-valued kernels, establishing conditions for solution regularity and existence.

Findings

Proved local and global existence of solutions under certain smoothness assumptions.

Established regularity properties for solutions of nonlocal wave equations.

Unified various physical wave models within a single mathematical framework.

Abstract

In this paper, the regularity properties of Cauchy problem for linear and nonlinear nonlocal wave equations are studied.The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform are operator functions defined in Hilbert space H together with some growth conditions. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data and the operator functions. By selecting the space H and the operators, the wide class of wave equations in the field of physics are obtained.