Differential Equations

, Volume 55, Issue 1, pp 60–67 | Cite as

Instantaneous Blow-Up of a Weak Solution of a Problem in Plasma Theory on the Half-Line

  • M. O. KorpusovEmail author
Partial Differential Equations


We consider a problem with some boundary and initial conditions for an equation arising in the theory of ion-sound waves in plasma. We prove that if the spatial (one-dimensional) variable ranges on an interval, then this problem has a unique nonextendable classical solution which in general exists only locally in time. If the spatial variable varies on the half-line, then, for the problem in question, we obtain an upper bound for the lifespan of its weak solution and find initial conditions for which there exist no solutions even locally in time (instantaneous blow-up of the weak solution). A similar result is obtained for the classical solution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gabov, S.A., Novye zadachi matematicheskoi teorii voln (New Problems of Mathematical Theory of Waves), Moscow: Nauka, 1998.zbMATHGoogle Scholar
  2. 2.
    Mitidieri, E. and Pokhozhaev, S.I., A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 2001, vol. 234, pp. 1–362.zbMATHGoogle Scholar
  3. 3.
    Bakhvalov, N.S., Zhileikin, Ya.M., and Zabolotskaya, E.A., Sovremennye problemy fiziki. Nelineinaya teoriya zvukovykh puchkov (Contemporary Problems of Physics. Nonlinear Theory of Sound Beams), Moscow: Nauka, 1982.Google Scholar
  4. 4.
    Korpusov, M.O. and Mikhailenko, S.G., Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation, Comput. Math. Math. Phys., 2017, vol. 57, no. 7, pp. 1167–1172.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Korpusov, M.O., Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type, Izv. Math., 2015, vol. 75, no. 5, pp. 955–1012.CrossRefzbMATHGoogle Scholar
  6. 6.
    Korpusov, M.O., Lukyanenko, D.V., Panin, A.A., and Yushkov, E.V., Blow-up for one Sobolev problem: theoretical approach and numerical analysis, J. Math. Anal. Appl., 2016, vol. 442, no. 2, pp. 451–468.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Panin, A.A., On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation, Math. Notes, 2015, vol. 97, no. 6, pp. 892–908.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations