On a priori estimates of the Kirchhoff equation integral load

Abstract


A large number of physical, biological and other phenomena are described by loaded equations. The nonlinear hyperbolic Kirchhoff equation models some oscillatory processes. It contains a load in the form of a rational degree m/n of a linear function of the norm of the desired solution in the space H1(Ω). We will call such a load an integral one. In this paper, a second mixed problem with homogeneous boundary conditions is considered for this equation. Due to the complexity of integrating nonlinear differential equations, in many cases they are approximated by linear equations with varying degrees of accuracy. In this case, it may turn out that the linearized equation very conditionally models the phenomenon under study.The purpose of this work is to establish a priori estimates for the integral load of the Kirchhoff equation. Subsequently, they are used for its "correct" linearization. The corresponding results are formulated in the form of theorems. In the case of a positive degree m/n, the obtained estimate is valid for any values of m and n. In the negative case, separate estimates are set for m < n, m = n and m > n. In all cases, a transition is made from the non-strict equality of the a priori assessment to equality. This equality relates the integral load to some linear function depending on the initial conditions and the right side of the equation. To reduce the Kirchhoff equation to a linear equation, its integral load is replaced by the resulting function. The method is applicable to equations with an integral load both in the main part and in the minor terms.

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About the authors

O. L Boziev

Kabardino-Balkarian State University; Institute of Computer Science and Problems of Regional Management of Kabardino-Balkarian Science Center of the Russian Academy of Sciences

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