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Abstract <p>For a fourth-order nonlinear partial differential equation in time that models the propagation of bending waves in a Timoshenko beam, the Cauchy problem is studied in the space of continuous functions defined on the entire number line and for which limits at infinity exist. The time interval for the existence and uniqueness of a classical solution to an auxiliary Cauchy problem related to the original problem is established, and an estimate for the norm of this local solution is given. Conditions are found that relate the local classical solutions of the original and auxiliary Cauchy problems over a certain time interval. Sufficient conditions for the extension of a local classical solution of the Cauchy problem to a global solution and for the blowup of the solution of the nonlinear Timoshenko equation over a finite time interval are considered.</p>

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Solution Blow-Up and Global Solvability of the Cauchy Problem for the Nonlinear Timoshenko Equation for Bending Oscillations of a Beam

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • Kh. G. Umarov

摘要

Abstract

For a fourth-order nonlinear partial differential equation in time that models the propagation of bending waves in a Timoshenko beam, the Cauchy problem is studied in the space of continuous functions defined on the entire number line and for which limits at infinity exist. The time interval for the existence and uniqueness of a classical solution to an auxiliary Cauchy problem related to the original problem is established, and an estimate for the norm of this local solution is given. Conditions are found that relate the local classical solutions of the original and auxiliary Cauchy problems over a certain time interval. Sufficient conditions for the extension of a local classical solution of the Cauchy problem to a global solution and for the blowup of the solution of the nonlinear Timoshenko equation over a finite time interval are considered.