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Blowup of Classical Solutions of Nonlinear Hyperbolic Equations: A Survey of Recent Results

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Partial Differential Equations and Mathematical Physics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 21))

Abstract

Some recent results on blowup of classical solutions to hyperbolic systems or equations are presented. Emphasis is put on the description of the mechanism, which can be given for some semilinear cases or for cases in one space dimension. For higher dimensions, only asymptotic results have been obtained so far.

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References

  1. Alinhac S., Explosion géométrique pour des systèmes quasilinéaires,Séminaire d’Équations aux dérivées partielles, Exp. no 1, École Polytechnique, Paris, (1993), and article to appear in Amer. J. Math. (1995).

    Google Scholar 

  2. Alinhac S., Temps de vie et comportement explosif des solutions d’équations d’ondes quasilinéaires en dimension deux II,Duke Math. J. 73 no 3 (1994), 543–560.

    Google Scholar 

  3. Alinhac S., Temps de vie précisé et explosion géométrique pour des systèmes hy-perboliques quasi-linéaires en dimension un d’espace,Séminaire d’Équations aux dérivées partielles, Exp. no 6, École Polytechnique, Paris, (1995), and article to appear in Annali di Pisa (1995).

    Google Scholar 

  4. Alinhac S., Blowup for nonlinear hyperbolic equations, Birkhäuser, 1995, Progress in Nonlinear Differential Equations and their Applications.

    MATH  Google Scholar 

  5. Caffarelli L. and Friedman A., The blowup boundary for nonlinear wave equations,Trans. Amer. Math. Soc. 297 no 1 (1986), 233–241.

    Google Scholar 

  6. Caffarelli L., Differentiability of the blowup curve for one dimensional nonlinear wave equations,Arch. Rat. Mech. Anal. 91 no 1 (1985), 83–98.

    Google Scholar 

  7. Christodoulou D., Bounded variation solutions of the spherically symmetric Einstein scalar field equations,Comm. Pure Appl. Math. 46 (1993), 1131–1220.

    Google Scholar 

  8. Hörmander L., The lifespan of classical solutions of nonlinear hyperbolic equations,Springer Lecture Notes in Math. 1256 (1986), 214–280.

    Google Scholar 

  9. John F., Nonlinear wave equations. Formation of singularities,Amer. Math. Soc., Providence, R. I., 1990, University Lecture Series 2, Lehigh University.

    Google Scholar 

  10. John F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405.

    Google Scholar 

  11. Joly J. L., Metivier G., Rauch J., A nonlinear instability for 3 x 3 systems of conservation laws, Preprint, Université de Bordeaux (1993).

    Google Scholar 

  12. Kichenassamy S. and Littman W., Blowup surfaces for nonlinear wave equations I and II,Comm. PDE 18 (1993), 431–452 and 1869–1899.

    Google Scholar 

  13. Kichenassamy S., The blow-up problem for exponential nonlinearities,Preprint (1995).

    Google Scholar 

  14. Klainerman S. and Majda A., Formation of singularities for wave equations including the nonlinear vibrating string,Comm. Pure Appl. Math. 33 (1980), 241–263.

    Google Scholar 

  15. Lax P. D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 no 5 (1964), 611–613.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lax P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537–566.

    MathSciNet  MATH  Google Scholar 

  17. Lebaud M. P., Description de la formation d’un choc dans le p-système, J. Math. Pure Appl. 73 (1994), 523–565.

    MathSciNet  MATH  Google Scholar 

  18. Lindblad H., Blowup of solutions of a, u—Du = luI’ with small initial data, Comm. PDE 15 no 6 (1990), 757–821.

    Google Scholar 

  19. Majda A., Compressible fluid flow and systems of conservation laws,Springer Verlag, 1984, Appl. Math. Sci. 53.

    Google Scholar 

  20. Natalini R., Unbounded solutions for conservation laws with sources,Nonlinear Anal. 21 no 5 (1993), 349–362.

    Google Scholar 

  21. Smoller J., Shock waves and reaction diffusion equations,Springer Verlag, New York, 1983, Grundlehr. d. Math. Wiss. 258.

    Google Scholar 

  22. Strauss W., Nonlinear wave equations, Conf. Board Math. Sc. 73 (1989).

    Google Scholar 

  23. Zuily C., Solutions en grand temps d’équations d’ondes non linéaires,Séminaire Bourbaki 779, Paris, (1993/1994).

    Google Scholar 

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Alinhac, S. (1996). Blowup of Classical Solutions of Nonlinear Hyperbolic Equations: A Survey of Recent Results. In: Hörmander, L., Melin, A. (eds) Partial Differential Equations and Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 21. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0775-7_2

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  • DOI: https://doi.org/10.1007/978-1-4612-0775-7_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6897-0

  • Online ISBN: 978-1-4612-0775-7

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