Theoretical and Mathematical Physics

, Volume 194, Issue 3, pp 347–359 | Cite as

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

Article
  • 9 Downloads

Abstract

We consider several nonlinear evolution equations sharing a nonlinearity of the form ∂2u2/∂t2. Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.

Keywords

finite-time blowup nonlinear wave instantaneous blowup 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media: Applications to Nonlinear Acoustics [in Russian], Fizmatlit, Moscow (2008); English transl.: Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics, Springer, Berlin (2011).MATHGoogle Scholar
  2. 2.
    M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Theory of Waves [in Russian], Nauka, Moscow (1990).MATHGoogle Scholar
  3. 3.
    N. S. Bahvalov, Ya. M. Zhileikin, and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams [in Russian], Nauka, Moscow (1982).Google Scholar
  4. 4.
    S. A. Gabov, New Problems in the Mathematical Theory of Waves [in Russian], Fizmatlit, Moscow (1998).MATHGoogle Scholar
  5. 5.
    E. Mitidieri and S. I. Pohozaev, “Nonexistence of weak solutions for some degenerate and singular hyperbolic problems on Rn+1 +,” Proc. Steklov Inst. Math., 232, 240–259 (2001).MATHGoogle Scholar
  6. 6.
    M. O. Korpusov and S. G. Mikhalenko, “Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation,” Comput. Math. Math. Phys., 57, 1167–1172 (2017).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. O. Korpusov, “Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type,” Izv. Math., 79, 955–1012 (2015).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations