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St. Petersburg Mathematical Journal

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On the destruction of ion-sound waves in plasma with strong space-time dispersion

Author: M. O. Korpusov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 23 (2011), nomer 6.
Journal: St. Petersburg Math. J. 23 (2012), 989-1011
MSC (2010): Primary 35Q60
Published electronically: September 17, 2012
MathSciNet review: 2962182
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Abstract: The object of study in this paper is a model equation that describes ion-sound waves in plasma with the account of strong nonlinear dissipation and nonlinear sources of general type, together with strong space-time dispersion. Sufficient destruction conditions are obtained for the solution of the corresponding initial boundary problem in a bounded three-dimensional domain with homogeneous Dirichlet-Neumann conditions on the boundary. Moreover, the life-time of the solution is estimated. Finally, it is proved that, for every initial data in $ \mathbb{H}_0^2(\Omega )$, a strong generalized solution of this problem exists locally in time, i.e., it is shown that the solution destruction time is always positive.

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  • 1. H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operator-differentialgleichungen, Math. Lehrbucher und Monogr., II Abteilung, Math. Monogr., Bd. 38, Akademie-Verlag, Berlin, 1974. MR 0636412 (58:30524a)
  • 2. V. A. Galaktionov and S. I. Pokhozhaev, Third-order nonlinear dispersion equations: shock waves, rarefaction waves, and blow-up waves, Zh. Vychisl. Mat. Mat. Fiz. 48 (2008), no. 10, 1819-1846; English transl., Comput. Math. Math Phys. 48 (2008), no. 10, 1784-1810. MR 2493769 (2009k:35031)
  • 3. V. K. Kalantarov and O. A. Ladyzhenskaya, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 77-102; English transl. in J. Soviet Math. 10 (1978), no. 1. MR 0604036 (58:29269)
  • 4. M. A. Krasnosel'skiĭ, Topological methods in the theory of nonlinear integral equations, Costekhizdat, Moscow, 1956; English transl., The Macmillan Co., New York, 1964. MR 0096983 (20:3464); MR 0159197 (28:2414)
  • 5. N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Nauchn. Kniga, Novosibirsk, 1998; Translated from the 1996 English original, Grad. Stud. in Math., vol. 12, Amer. Math. Soc., Providence, RI, 1996. MR 1657890 (99g:35002); MR 1406091 (97i:35001)
  • 6. L. D. Landau and E. M. Lifshitz, Theoretical physics. Vol. 8. Electrodynamics of continuous media, 3rd ed., Nauka, Moscow, 1992; English transl., Pergamon Press, Oxford, 1960. MR 1330694 (95m:00001); MR 0121049 (22:11796)
  • 7. -, Theoretical physics. Vol. 10. Physical kinetics, Fizmatlit, Moscow, 2001. (Russian)
  • 8. È. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1-384; English transl., Proc. Steklov Inst. Math. 2001, no. 3 (234), 1-362. MR 1879326 (2005d:35004)
  • 9. A. A. Samarskiĭ, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaĭlov, Peaking modes in problems for quasilinear parabolic equations, Nauka, Moscow, 1987; English transl., Blow-up in quasilinear parabolic equations, Gruyter Exp. in Math., vol. 19, Walter de Gruyter, Berlin, 1995. MR 0919951 (89a:35002); MR 1330922 (96b:35003)
  • 10. A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov, and Yu. D. Pletner, Linear and nonlinear equations of the Sobolev type, Fizmatlit, 2007, 734 pp. (Russian)
  • 11. I. E. Tamm, Foundations of the electricity theory, Nauka, Moscow, 1989. (Russian)
  • 12. L. Gasinski and N. S. Papageorgiou, Nonlinear analysis, Ser. in Math. Anal. Appl., vol. 9, Chapman and Hall, Boca Raton, FL, 2006. MR 2168068 (2006e:47001)
  • 13. H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $ Pu_t=-Au+\mathcal {F}(u)$, Arch. Rational Mech. Anal. 51 (1973), 371-386. MR 0348216 (50:714)
  • 14. -, Instability and nonexistence of global solutions to nonlinear wave equations of the form $ Pu_{tt}=-Au+\mathcal {F}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1-21. MR 0344697 (49:9436)

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Additional Information

M. O. Korpusov
Affiliation: Department of Physics, Lomonosov Moscow State University, Moscow, Russia

Keywords: Destruction, equations of Sobolev type, nonlinear analysis
Received by editor(s): May 27, 2010
Published electronically: September 17, 2012
Additional Notes: Supported by RFBR, grant no. 11-01-12018-ofi-m-2011, and by the President program for the support of young doctors of science, MD-99.2009.1
Article copyright: © Copyright 2012 American Mathematical Society

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