Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients

Ferruccio Colombini; Daniele del Santo; Tamotu Kinoshita

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 2, page 327-358
  • ISSN: 0391-173X

Abstract

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We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in 𝒞 ∞ and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.

How to cite

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Colombini, Ferruccio, del Santo, Daniele, and Kinoshita, Tamotu. "Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.2 (2002): 327-358. <http://eudml.org/doc/84473>.

@article{Colombini2002,
abstract = {We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in $\{\mathcal \{C\}\}^\infty $ and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.},
author = {Colombini, Ferruccio, del Santo, Daniele, Kinoshita, Tamotu},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {strict hyperbolicity},
language = {eng},
number = {2},
pages = {327-358},
publisher = {Scuola normale superiore},
title = {Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients},
url = {http://eudml.org/doc/84473},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Colombini, Ferruccio
AU - del Santo, Daniele
AU - Kinoshita, Tamotu
TI - Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 2
SP - 327
EP - 358
AB - We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in ${\mathcal {C}}^\infty $ and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.
LA - eng
KW - strict hyperbolicity
UR - http://eudml.org/doc/84473
ER -

References

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  1. [1] F. Colombini – E. De Giorgi – S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temp, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (1979), 511-559. Zbl0417.35049MR553796
  2. [2] F. Colombini – D. Del Santo – T. Kinoshita, On the Cauchy problem for hyperbolic operators with non-regular coefficients, to appear in Proceedings of the Conference “À la mémoire de Jean Leray” Karlskrona 2000, M. de Gosson – J. Vaillant (eds.), Kluwer, New York. Zbl1036.35122MR2051477
  3. [3] F. Colombini – N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657-698. Zbl0840.35067MR1324638
  4. [4] F. Colombini – S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann. Sci. École Norm. Sup. (4) 22 (1989), 109-125. Zbl0702.35146MR985857
  5. [5] L. Hörmander, “Linear Partial Differential Operators”, Springer-Verlag, Berlin, 1963. Zbl0108.09301
  6. [6] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl. 140 (1985), 133-145. Zbl0583.35074MR807634
  7. [7] T. Nishitani, Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x , Bull. Sci. Math. 107 (1983), 113-138. Zbl0536.35042MR704720

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