Curvilinear parallelogram identity and mean-value property for a semilinear hyperbolic equation of second-order

TL;DR

This paper explores the curvilinear parallelogram identity and mean-value property for solutions of a class of second-order hyperbolic equations, providing insights into their qualitative behavior and solution methods.

Contribution

It introduces the curvilinear parallelogram identity and mean-value property for second-order hyperbolic equations with specific coefficient conditions, advancing understanding of their solution properties.

Findings

Identifies the curvilinear parallelogram identity for certain hyperbolic equations

Establishes the mean-value property for solutions of these equations

Provides potential methods for solving initial-boundary value problems

Abstract

In this paper, we discuss some of the important qualitative properties of solutions of second-order hyperbolic equations, whose coefficients of the terms involving the second-order derivatives are independent of the desired function and its derivatives. Solutions of these equations have a special property called the curvilinear parallelogram identity (or the mean-value property), which can be used to solve some initial-boundary value problems.