Representation of Classical Solutions to a Linear Wave Equation with Pure Delay

TL;DR

This paper investigates a wave equation with pure delay in a 1D bounded domain, proving the existence, uniqueness, and explicit representation of classical solutions, along with their continuous dependence on initial and boundary data.

Contribution

It provides a rigorous analysis of classical solutions for delayed wave equations, including explicit formulas and stability results, which were previously lacking.

Findings

Unique existence of classical solutions for any finite time horizon.

Explicit representation formulas for solutions.

Continuous dependence on data in a weak norm.

Abstract

For a wave equation with pure delay, we study an inhomogeneous initial-boundary value problem in a bounded 1D domain. Under smoothness assumptions, we prove unique existence of classical solutions for any given finite time horizon and give their explicit representation. Continuous dependence on the data in a weak extrapolated norm is also shown.