A priori estimates of the integral load of the hyperbolic Kirchhoff equation

Aim . The aim of this work is to establish a priori estimates for the integral load of the Kirchhoff equation. This equation models some nonlinear oscillatory processes. Here, the load is the rational degree m / n of a linear function of the norm of the desired solution in the space H 1 (Ω). Methodology . To establish a priori estimates, integral transformations of the terms of the scalar product of the original equation and the time derivative of its solution are performed. The application of some well-known integral inequality leads to the desired estimates. Results . A priori inequalities limiting the integral load of the Kirchhoff equation to a known function are established. This function depends on the right side of the equation and the initial conditions, depending on the sign and type of exponent. A method is shown for reducing the Kirchhoff equation to a linear equation by replacing the integral load with the right-hand sides of these estimates. An example of such a reduction is given. Research implications . The described method of establishing a priori estimates and subsequent reduction of a nonlinear equation to a linear one can be applied to a wide class of loaded equations containing the modulus of the integral of the rational degree of the desired function or its derivative.