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 H1(Ω). 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.