Abstract: For the differential equation mentioned in the title of the article, the solvability of the Cauchy problem in the space of continuous functions on the whole real axis by reducing to an abstract Cauchy problem in a Banach space is studied. An explicit form of the solution of the corresponding linear equation is found. The time interval for the existence of the classical solution of the Cauchy problem for a nonlinear equation is established and an estimate of the norm of this local solution is obtained. The conditions for the existence of a global solution and the destruction of the solution on a finite interval are considered.
Keywords: bending vibrations of a rod, Klein-Gordon equation, strongly continuous semigroups of operators
For citation: Umarov Kh. G. The Cauchy problem for the equation of bending vibrations of a
nonlinear-elastic rod of infinite length // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 59-69.
DOI 10.23671/VNC.2017.3.7131
1. Ostrovskij L. A., Potapov A. I. Vvedenie v teoriju modulirovannyh
voln[Introduction to the Theory of Modulated Waves]. M.: Fizmatlit,
2003. 400 p.
2. Erofeev V. I., Kazhaev V. V., Semerikova N. P. Volny v
sterzhnjah. Dispersija. Dissipacija. Nelinejnost' [Waves in rods.
Dispersion. Dissipation. Nonlinearity]. M.: Fizmatlit, 2002. 208 p.
3. Dunford N., Schwartz J. T. Linear Operators. P. I: General
Theory. N.Y.: Interscience, 1958. 872 p.
4. Krein S. G. Linear Differential Equations in Banach Space.
Providence (RI): Amer. Math. Soc., 1971. 390 p.
5. Vasil'ev V. V., Krejn S. G., Piskarev S. I. Semigroups of operators,
cosine of operator-functions and linear differential equations. Itogi nauki i tehniki. Serija
Matematicheskij analiz [Results of science and technology. Ser.
Math. Anal.]. VINITI, 1990. vol. 28, pp. 87-202.
6. Prudnikov A. P., Brychkov Yu. A., Marichev O. I. Integrals and
Series, vol. 2, Special Functions, Gordon & Breach Sci. Publ., New
York, 1990. 750 p.
7. Prudnikov A. P., Brychkov Yu. A., Marichev O. I. Integrals and
Series, Vol. 3, More Special Functions, Gordon & Breach Sci. Publ.,
New York, 1990. 800 p.
8. Krasnoselskii M. A., Zabreiko R. P., Pustylnik E. I. Sobolevskii
P. E. Integral Operators in Spaces of Summable Functions. Leyden:
Noordhoff, 1976. 520 p.
9. Travis C. C., Webb G. F. Cosine families and abstract nonlinear
second order differential equations. Acta Mathematica Academiae
Scientiarum Hungaricae, 1978, 32, pp. 75–96.
10. Appell J., Zabreiko P. P. Nonlinear Superposition Operators.
Cambridge: Cambridge Univ. Press, 1990. 320 p.
11. Dragomir S. S. Some Gronwall Type Inequalities and Applications.
Melbourne City MC, 2002. 193 p.
12. Benjamin T. B., Bona J. L., Mahony J. J., Model equations for
long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc.
London, 1972, vol. 272, pp. 47–78.
13. Korpusov M. O., Sveshnikov A. G., Jushkov E. V. Metody teorii
razrushenija reshenij nelinejnyh uravnenij matematicheskoj fiziki.
[Methods of the theory of fracture of solutions of nonlinear
equations of mathematical physics]. M.: Fizicheskij fakul'tet MGU,
2014. 364 p.