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The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types

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Abstract

Collapse in finite time is established for part of the solutions of certain classes of quasilinear equations of parabolic and hyperbolic types, the linear part of which has general form. Certain hyperbolic equations having L-M pairs belong to these classes.

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Literature cited

  1. S. N. Bernshtein, Collected Works, Vol. III, Izd. Akad. Nauk SSSR, Moscow (1960).

    Google Scholar 

  2. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).

  3. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Types, Amer. Math. Soc., Providence, Rhode Island (1968).

    Google Scholar 

  4. S. Kaplan, “On the growth of solutions of quasilinear parabolic equations,” Comm. Pure Appl. Math.,16, 305–330 (1963).

    Google Scholar 

  5. J. B. Keller, “On solutions of nonlinear wave equations,” Comm. Pure App. Math.,10, 523–530 (1957).

    Google Scholar 

  6. H. Fujita, “On the blowing up of solutions of the Cauchy problem for ut=Δu+u1+d,” J. Faculty Sci., Univ. Tokyo,13, 109–124 (1969).

    Google Scholar 

  7. H. Fujita, “On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations,” Proc. Symp. Pure Math., Nonlinear Functional Analysis, Vol. 18, Amer. Math. Soc., Providence, Rhode Island (1968), pp. 105–113.

    Google Scholar 

  8. R. T. Glassey, “Blow-up theorems of nonlinear wave equations,” Math. Z.,132, 183–203 (1973).

    Google Scholar 

  9. D. H. Sattinger, “Stability of nonlinear hyperbolic equations,” Arch. Rat. Mech. Anal.,28, 226–244 (1967).

    Google Scholar 

  10. N. J. Zabusky, “Exact solution for the vibrations of a nonlinear continuous mode string,” J. Math. Phys.,3, 1028–1039 (1962).

    Google Scholar 

  11. P. D. Lax, “Development of singularities of solutions of nonlinear hyperbolic partial differential equations,” J. Math. Phys.,5, 611–613 (1964).

    Google Scholar 

  12. R. C. MacCamy and V. J. Mizel, “Existence and nonexistence in the large of solutions of quasilinear wave equations,” Arch. Rat. Mech. Anal.,25, 298–320 (1967).

    Google Scholar 

  13. A. B. Shabat, “On the Cauchy problem for the Ginzburg-Landau equation,” Dynamics of a Continuous Medium, Vol. 1, Novosibirsk (1969), pp. 180–194.

    Google Scholar 

  14. V. N. Vlasov, V. I. Talanov, and V. A. Petrishchev, “Averaged description of wave packets in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.14, 1353–1363 (1971).

    Google Scholar 

  15. L. M. Legtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of the collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz.,68, 115–126 (1975).

    Google Scholar 

  16. A. V. Zhiber, “The Cauchy problem for a class of semilinear systems of differential equations,” Doctoral Dissertation, Moscow (1975).

  17. H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=−Au+ℱ(u),” Trans. Am. Math. Soc.,192, 1–21 (1974).

    Google Scholar 

  18. H. A. Levine, “Some nonexistence and instability theorems for formally parabolic equations of the form Put=−Au+ℱ(u),” Arch. Rat. Mech. Anal.,51, 371–386 (1973).

    Google Scholar 

  19. R. I. Knops, H. A. Levine, and L. E. Payne, “Nonexistence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics,” Arch. Rat. Mech. Anal.,55, 52–72 (1974).

    Google Scholar 

  20. B. Straughan, “Further nonexistence theorems for abstract nonlinear wave equations,” Proc. Am. Math. Soc.,48, 381–390 (1975).

    Google Scholar 

  21. H. Flaschka, “Toda lattice. I,” Phys. Rev.,B9, 1921–1923 (1974).

    Google Scholar 

  22. H. Flaschka, “Toda lattice. II,” Progr. Theor. Phys.,51, 703–716 (1974).

    Google Scholar 

  23. S. V. Manakov, “On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, 543–555 (1974).

    Google Scholar 

  24. J. Moser, “Finitely many mass points on the line under the influence of an exponential potential. An integrable system,” Preprint (1974).

  25. V. E. Zakharov and S. V. Manakov, “On the resonance interaction of wave packets in non-linear media,” Pis'ma Zh. Eksp. Teor. Fiz.,18, 413–417 (1973).

    Google Scholar 

  26. M. J. Ablowitz, D. S. Kaup, A. C. Newel, and H. Segur, “Method for solving the sine-Gordon equation,” Phys. Rev. Lett.,30, 1262 (1973).

    Google Scholar 

  27. L. A. Takhtadzhyan and L. D. Faddeev, “An essentially nonlinear one-dimensional model of classical field theory,” Teor. Mat. Fiz'.,21, 160–174 (1974).

    Google Scholar 

  28. V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, “A complete description of solutions of the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,219, 1334–1337 (1974).

    Google Scholar 

  29. V. E. Zakharov, “On the problem of stochastization of one-dimensional chains of non-linear operators,” Zh. Eksp. Teor. Fiz.,65, 219–228 (1973).

    Google Scholar 

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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 77–102, 1977.

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Kalantarov, V.K., Ladyzhenskaya, O.A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J Math Sci 10, 53–70 (1978). https://doi.org/10.1007/BF01109723

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