Skip to main content
Springer Nature Link
Log in

Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We obtain conditions for the existence of a global solution and the blow-up of the solution of the Cauchy problem on a finite time interval for a nonlinear third-order partial differential equation generalizing the equation of torsion vibrations of a cylindrical rod with allowance for internal and external damping modeling the propagation of longitudinal stress waves along a one-dimensional viscoelastic rod whose the material obeys Voigt–Kelvin medium deformation law.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Notes

  1. A classical solution is understood as a sufficiently smooth function that has all continuous derivatives of the desired order and satisfies the equation at each point in the domain where it is given.

  2. Note that \(\beta \xi -2\sqrt {\alpha \delta }|\xi |=2\sqrt {\alpha \delta }|\xi |K(\xi )<0\), where \( K(\xi )=\mathrm {sign}\thinspace (\xi ){\beta }/{(2\sqrt {\alpha \delta })}-1 \), \(\xi \in \mathbb {R}^1 \).

  3. The elements of the Banach space \(C[\mathbb {R}^1] \) satisfy the estimate \({\|\varphi \cdot \psi \|}_C\le {\|\varphi \|}_C {\|\psi \|}_C\); this makes it possible to operate with it like with an algebra [11, Sec. 6.1].

  4. Here, taking into account the a priori estimates (50) and (57), the following algorithm is used [12]: we take \(v(t_0,x) \) as a new initial function and note that the function \(v(t_0,x) \) has the same a set of properties that, being assumed for the function \(v(0,x)\), made it possible to prove the existence of both the classical solution \(v(t,x) \) of Eq. (4) on the interval \(0\le t\le t_0 \) and the corresponding solution \(u(t,x) \) to Eq. (1) on the same interval \([0,t_0] \), the classical solution \(v(t,x) \) extends from the interval \([0,t_0] \) to the classical solution \(v(t,x) \), \(t\in [0,t_0+t_1] \), where the value of \(t_1 \) depends on the parameters \(\alpha \), \(\beta \), and \(\gamma \), the initial functions \(\varphi ( x) \) and \(\psi (x) \), and the nonlinearity \(\sigma (\cdot ) \). Repeating this process sufficiently many times, we obtain an unbounded extension of the time interval on which the solution is guaranteed; i.e., we obtain a classical solution of the Cauchy problem for Eq. (1) on an arbitrary time interval.

REFERENCES

  1. Dunford, N. and Schwartz, J.T., Linear Operators. Part I: General Theory, New York–London: Interscience, 1958. Translated under the title: Lineinye operatory. Obshchaya teoriya, Moscow: Izd. Inostr. Lit., 1962.

    MATH  Google Scholar 

  2. Filippov, A.P., Kolebaniya deformiruemykh sistem (Oscillations in Deformable Systems), Moscow: Mashinostroenie, 1970.

    Google Scholar 

  3. Erofeev, V.I., Kazhaev, V.V., and Semerikova, N.P., Volny v sterzhnyakh. Dispersiya. Dissipatsiya. Nelineinost’ (Waves in Rods. Dispersion. Dissipation. Nonlinearity), Moscow: Fizmatlit, 2002.

    Google Scholar 

  4. Greenberg, J.M., McCamy, R.C., and Mizel, V.J., On the existence, uniqueness and stability of solutions of the equation \(\sigma ^{\prime }(u_x)u_{xx}+\lambda {u}_{xtx}=\rho _0u_{tt} \), J. Math. Mech., 1968, vol. 17, pp. 707–728.

    MathSciNet  Google Scholar 

  5. Webb, G.F., Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Can. J. Math., 1980, vol. 32, no. 3, pp. 631–643.

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrews, G., On the existence of solutions to the equation \(u_{tt}=u_{xxt}+\sigma (u_x)_x \), J. Differ. Equat., 1980, vol. 35, no. 2, pp. 200–231.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kozhanov, A.I., Lar’kin, N.A., and Yanenko, N.N., A mixed problem for a class of equations of third order, Sib. Math. J., 1982, vol. 22, no. 6, pp. 867–872.

    Article  MATH  Google Scholar 

  8. Lar’kin, N.A., Novikov, V.A., and Yanenko, H.H., Nelineinye uravneniya peremennogo tipa (Nonlinear Equations of Variable Type), Novosibirsk: Nauka, 1983.

    MATH  Google Scholar 

  9. Vasil’ev, V.V., Krein, S.G., and Piskarev, S.I., Semigroups of operators, cosine operator functions, and linear differential equations, J. Sov. Math., 1991, vol. 54, no. 4, pp. 1042–1129.

    Article  MATH  Google Scholar 

  10. Dragomir, S.S., Some Gronwall Type Inequalities and Applications, Melbourne: Nova Science, 2002.

    Google Scholar 

  11. Appell, J. and Zabreiko, P.P., Nonlinear Superposition Operators, Cambridge: Cambridge Univ. Press, 1990.

  12. Benjamin, T.B., Bona, J.L., and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. London, 1972, vol. 272, pp. 47–78.

    Article  MathSciNet  MATH  Google Scholar 

  13. Korpusov, M.O., Sveshnikov, A.G., and Yushkov, E.V., Metody teorii razrusheniya reshenii nelineinykh uravnenii matematicheskoi fiziki (Methods of the Theory of Blow-Up of Solutions to Nonlinear Equations of Mathematical Physics), Moscow: Izd. Mosk. Gos. Univ., 2014.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Sciences of Chechen Republic, Groznyi, 364024, Russia

    Kh. G. Umarov

  2. Chechen State Pedagogical University, Groznyi, 364068, Russia

    Kh. G. Umarov

Authors
  1. Kh. G. Umarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kh. G. Umarov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Umarov, K.G. Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation. Diff Equat 59, 51–73 (2023). https://doi.org/10.1134/S0012266123010056

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266123010056

Search

Navigation