Application of a priori estimates of the integral load of the Kirchhoff hyperbolic equation for its reduction to a linear equation

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 $H^1(\Omega)$. To achieve the specified goal, 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 Gronwall-Bellman type integral inequality leads to the desired estimates. A priori inequalities limiting the integral load of the Kirchhoff equation to a known function are established. This function depends on the right-hand side of the equation and the initial conditions, as well as on the sign and type of the exponent. The article shows a method 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. 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.