A priori estimates of the integral load of the hyperbolic Kirchhoff equation
https://doi.org/10.18384/2949-5067-2024-4-26-36
Abstract
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 H1(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.
About the Author
O. L. Boziev
Kabardino-Balkarian State University named after H. M. Berbekov; Institute of Computer Science and Problems of Regional Management, Kabardin-Balkar Scientific Center of Russian Academy of Sciences
Russian Federation
Russian Federation
Oleg L. Boziev – Cand. Sci (Phys.-Math.), Assoc. Prof., Department of Computer Technologies and Information Security, Institute of Artificial Intelligence and Digital Technologies, Kabardino-Balkarian State University named after H. M. Berbekov; Senior Researcher, Department of Automation and Informatization of Regional Control Systems, Institute of Computer Science and Problems of Regional Management, Kabardin-Balkar Scientific Center of Russian Academy of Sciences
73, Nalchik 360004, Kabardino-Balkarian Republic
ulitsa I. Armand 37A, Nalchik 360000, Kabardino-Balkarian Republic
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