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Energy Methods in Continuum Mechanics pp 1–12Cite as

Quasilinear Parabolic Equations with Non-Isotropic Nonlinearities: Space and Time Localization

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Abstract

The energy method is used to study some support properties of local solutions of second-order nonlinear parabolic equations with non-isotropic nonlinearities with respect to the solution and its spatial derivatives. We establish such properties of local weak solutions as finite speed of propagations of the initial disturbances, the waiting time phenomenon, and stable localization. The conditions providing these effects are formulated in terms of local assumptions on the data and the character of nonlinearity of the equation under consideration.

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References

  1. S.N.Antontsev On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. Dokl. Akad. Nauk. SSSR, 1981, 260, n.6, pp.1289–1293. (in Russian; English translation in Soviet Math. Dokl., 1981, 24, n. 2, pp. 420–424 ).

    MathSciNet  Google Scholar 

  2. S.N.Antontsev Localization of solutions of degenerate equations of continuum mechanics. Akad. Nauk. SSSR, Sibirsk. Otdel. Inst. Hydrodynamics, 1986, 109 pp. (Russian)

    Google Scholar 

  3. S. N. Antontsev and J. I. Diaz New results on localization of solutions of nonolinear elliptic and parabolic equations obtained by the energy method. Dokl. Akad.N auk SSSR. 1988, Vol. 303, No.3, pp. 320–325.(English translation in Soviet Math. Dokl. 1988, Vol.33, No.3, pp.535–539).

    Google Scholar 

  4. S.N.Antontsev, J.I.Diaz and S.I.Shmarev New applications of energy methods to parabolic free boundary Problems. Uspekhi Mat. Nauk, 1991, 46, issue 6 (282), pp. 181–182. (in Russian; English translation: in Russian Mathematical Surveys, 1991, 46, n. 6, pp. 193–194 ).

    Google Scholar 

  5. S.N.Antontsev, J.I.Diaz and S.I.Shmarev New results of the character of localization of solutions for elliptic and parabolic equations. Birkhauser-Verlag, International Series of Numerical Mathematics, 1992, 106, pp. 59–66.

    MathSciNet  Google Scholar 

  6. S.N.Antontsev, J.I.Diaz and S.I.Shmarev The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms. (To appear).

    Google Scholar 

  7. S.N.Antontsev, S.I. Shmarev Localization of solutions of nonlinear parabolic equations with linear sources of general type. Dinamica Sploshnoi Sredy, Novosibirsk, 1989, 89, pp. 28–42.

    MathSciNet  MATH  Google Scholar 

  8. S.N.Antontsev, S.I.Shmarev Local energy method and vanishing of weak solutions of nonlinear parabolic equations. Dokl. Akad. Nauk SSSR, 1991, 318, n.4. pp. 777–781. (in Russian; English translation: in Soviet Math. Doklady, 1991, 43 ).

    MathSciNet  Google Scholar 

  9. S.N.Antontsev, S.I.Shmarev Local energy method and vanishing properties of weak solutions of quasilinear parabolic equations. In book: Free boundary problems involving solids. Proceedings of the International Colloquim ” Free Boundary Problems: Theory and Applications”. J.Chadam and H.Rasmussen eds., Pitman Res. Notes in Math. ser., 281, Longman. 1993, pp. 2–5.

    Google Scholar 

  10. J.I. Diaz Nonlinear partial differential equations and free boundaries. Vol.1. Elliptic Equations. Research Notes in Math. No. 106. Pitman. 1985.

    MATH  Google Scholar 

  11. J.I.Diaz, L.Veron Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc., 1985, 290, n. 2, pp. 787–814.

    Article  MathSciNet  MATH  Google Scholar 

  12. A.S.Kalashnikov Some questions of the qualitative theory of nonlinear degenerate second-order parabolic equations. Russian Math Surveys, 1987, V. 42, n. 2, pp. 169–222.

    Article  MathSciNet  MATH  Google Scholar 

  13. O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva Linear and quasilinear equations of parabolic type. Trans. Math. Monographs. 23. Amer. Math. Soc. Providence. 1968.

    Google Scholar 

  14. J.L. Lions Some methods of solutions of nonlinear boundary value problems. Moscow. Mir. 1972. p. 288.

    Google Scholar 

  15. A.A.Samarsky, V.A.Galaktionov, S.P.Kurdumov, A.M.Mikhaylov Blow-up in Problems for Quasilinear Parabolic Problems, Nauka, Moscow, 1987 (in Russian; English translation: Walter de Groyter, Berlin, to appear).

    Google Scholar 

  16. S.I.Shmarev Local properties of solutions of nonlinear equation of heat conduction with absoption. Dinamika Sploshnoi Sredy, Novosibirsk, 1990, 95, pp. 28–42. (in Russian)

    Google Scholar 

  17. S.I.Shmarev The effect of support shrinking for solutions of a degenerate parabolic equation. Dinamica sploshnoi Sredy, 1990, 97, pp. 194–200. (in Russian)

    MathSciNet  MATH  Google Scholar 

  18. Yu. G. Rykov On the finite velocity of propagation of perturbation and inertia for generalized soluitions of degenerating quasilinear equations. Differential Equations and Their Applications, Moscow, Gos. Univ. Moscow, 1984, pp. 160–164.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Lavrentyev Prospect 15, Novosibirsk, 630090, Russia

    S. N. Antontsev

  2. University of Oviedo, c/Calvo Sotelo, s/n, Oviedo, 33007, Spain

    S. N. Antontsev

Authors
  1. S. N. Antontsev
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Editors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia

    S. N. Antontsev  & S. I. Shmarev  & 

  2. Universidad Complutense de Madrid, Spain

    J. I. Díaz

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© 1996 Kluwer Academic Publishers

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Antontsev, S.N. (1996). Quasilinear Parabolic Equations with Non-Isotropic Nonlinearities: Space and Time Localization. In: Antontsev, S.N., Díaz, J.I., Shmarev, S.I. (eds) Energy Methods in Continuum Mechanics. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0337-1_1

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  • DOI: https://doi.org/10.1007/978-94-009-0337-1_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6638-9

  • Online ISBN: 978-94-009-0337-1

  • eBook Packages: Springer Book Archive

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