Blow-up of the Solution to the Cauchy Problem for the Equation of Small Oscillations of an Absolutely Flexible Homogeneous Rod

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/c8669-9426-6840-b Blow-up of the Solution to the Cauchy Problem for the Equation of Small Oscillations of an Absolutely Flexible Homogeneous Rod Umarov, Kh. G. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3. Abstract: The Cauchy problem is studied for a nonlinear partial differential equation of Sobolev type (i.\,e., an equation not solved for the highest time derivative) that generalizes the equation of small oscillations of an absolutely flexible homogeneous rod. The problem is considered in the space of continuous functions defined on the entire real axis, for which limits exist at infinity. An explicit form of the classical solution to the corresponding linear homogeneous equation is obtained, and norm estimates for the operator-valued functions representing this solution are derived. An estimate for the norm of the solution to the Cauchy problem for the linear homogeneous equation is also given. The time interval of existence and uniqueness of a classical solution to the auxiliary Cauchy problem related to the original one is established, along with an estimate for the norm of this local solution. Conditions are found that ensure a correspondence between the classical solutions of the original and auxiliary Cauchy problems on a certain time interval. Moreover, conditions are examined under which the classical solution to the Cauchy problem blows up on a finite time interval. Keywords: small oscillations of an absolutely flexible homogeneous rod, nonlinear Sobolev-type equation, blow-up of solution Language: English Download the full text Download JATS XML For citation: Umarov, Kh.G. Blow-up of the Solution to the Cauchy Problem for the Equation of Small Oscillations of an Absolutely Flexible Homogeneous Rod, Vladikavkaz Math. J., 2025, vol. 28, no. 3, pp.121-133. DOI 10.46698/c8669-9426-6840-b + References 1. Demidenko, G. V., Uspensky, S. V. Equations and Systems not Resolved with Respect to the Highest Derivative, Novosibirsk, Scientific Book, 1998, 436 p. 2. Svetlitsky, V. A. Mechanics of Flexible Rods and Threads, Moscow, Mashinostroenie, 1978, 222 p. 3. Sveshnikov, A. G., Alshin, A. B., Korpusov, M. O. and Pletner, Yu. D. Linear and Nonlinear Equations of Sobolev Type, Moscow, Fizmatlit, 2007, 736 p. 4. Dunford, N. and Schwartz, J. T. Linear Operators. General Theory, Moscow, Foreign Literature Publishing House, 1962, 895 p. 5. Erofeev, I. V. Bending-Torsional, Longitudinal-Bending and Longitudinal-Torsional Waves in Rods, Bulletin of Scientific and Technical Development, 2012, vol. 5, no. 57, pp. 3–18. 6. Demidenko, G. V. Solvability Conditions of the Cauchy Problem for Pseudohyperbolic Equations, Siberian Mathematical Journal, 2015, vol. 56, no. 6, pp. 1028–1041. DOI: 10.1134/S0037446615060075. 7. Umarov, Kh. G. Cauchy Problem for the Equation of Bending Vibrations of a Nonlinearly Elastic Rod of Infinite Length, Vladikavkaz Mathematical Journal, 2017, vol. 19, no. 3, pp. 59–69. DOI: 10.23671/VNC.2017.3.7131. 8. Vasiliev, V. V., Krein, S. G. and Piskarev, S. I. Operator Semigroups, Cosine Operator Functions and Linear Differential Equations, Journal of Soviet Mathematics, 1991, vol. 54, no. 4, pp. 1042–1129. DOI: 10.1007/BF01138948. 9. Travis, C. C. and Webb, G. F. Cosine Families and Abstract Nonlinear Second Order Differential Equations, Acta Mathematica Academiae Scientiarum Hungaricae, 1978, vol. 32, no. 1–2, pp. 75–96. DOI: 10.1007/BF01902205. 10. Dragomir, S. S. Some Gronwall Type Inequalities and Applications, Melbourne City MC, 2002, 193 p. 11. Kudryavtsev, L. D. Course of Mathematical Analysis, Vol. \(1\), Moscow, Drofa, 2006, 702 p. 12. Korpusov, M. O., Sveshnikov, A. G. and Yushkov, E. V. Methods of the Theory of Destruction of Solutions of Nonlinear Equations of Mathematical Physics, Moscow, Physics Faculty of Moscow State University, 2014, 364 p. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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