Conditions for the existence of an "isolated" solution of a boundary value problem for a semilinear loaded hyperbolic equation

Boundary value problems for hyperbolic equations are an important area of mathematical physics and science in nature. They arise in various physical and engineering contexts and have a wide range of applications, including wave propagation in elastic media, electromagnetic waves, as well as problems related to fluid and gas motion. In this article, we will focus on one of the significant subclasses of hyperbolic equations, namely, semi-linear loaded hyperbolic equations, and examine the conditions for the existence of isolated solutions to boundary value problems for such equations.Semi-linear loaded hyperbolic equations are equations in which nonlinear terms depend on the solutions themselves. This makes their study more complex and mathematically intriguing. Our task is to find conditions under which such equations have isolated solutions, meaning solutions that exist in a bounded region of space and time and remain bounded themselves.Studying the conditions for the existence of isolated solutions for semi-linear loaded hyperbolic equations is of significant importance both in theory and practical applications. In this article, we will explore various approaches and methods used to analyze. In [1], issues related to loaded equations and their applications are investigated. The computational method for solving boundary value problems for loaded integro-differential equations and the correct solvability of boundary value problems for loaded differential equations were studied in works [2],[3]. Various problems for loaded differential equations and methods for finding their solutions are considered in [4-9].