The study of the existence and the analysis of the Cauchy problem solutionfor the perturbed Klein-Gordon equation

Background. At present time different approximate models for the description of the gas-liquid mix move are used, and the Klein – Gordon equation is one of them. Studying acoustic properties of liquids with bubbles of gas, as well as waves with the finite length in mixes with sufficiently large bubbles is based on these models. Besides, there are a number of mathematical models describing nonlinear seismic effects in geophysical environments, for instance, the sine-Gordon equation and its modifications. The objective of the paper is to study the existence and the Cauchy problem solution for the perturbed Klein – Gordon equation and to determine the relative error of the solution of Cauchy problem when replacing the perturbed Klein – Gordon equation with the non-perturbed one. Results and conclusions. The existence of the Cauchy problem solution for the perturbed Klein-Gordon equation has been proved. The relative error of the solution of the Cauchy problem for the perturbed Klein – Gordon equation when replacing the perturbed Klein – Gordon equation with non-perturbed has been determined. The results of the research make it possible to find borders at which it is admissible to replace small constant coefficients with zero.