Theoretical and Mathematical Physics

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

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

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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 
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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

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