The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable

Kunihiko Kajitani; Yasuo Yuzawa

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 4, page 465-482
  • ISSN: 0391-173X

Abstract

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We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.

How to cite

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Kajitani, Kunihiko, and Yuzawa, Yasuo. "The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.4 (2006): 465-482. <http://eudml.org/doc/242445>.

@article{Kajitani2006,
abstract = {We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.},
author = {Kajitani, Kunihiko, Yuzawa, Yasuo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {465-482},
publisher = {Scuola Normale Superiore, Pisa},
title = {The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable},
url = {http://eudml.org/doc/242445},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Kajitani, Kunihiko
AU - Yuzawa, Yasuo
TI - The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 4
SP - 465
EP - 482
AB - We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski’s method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.
LA - eng
UR - http://eudml.org/doc/242445
ER -

References

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  2. [2] F. Colombini, E. Jannelli and S. Spagnolo, Wellposedness in the Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983), 291–312. Zbl0543.35056MR728438
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  4. [4] K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J. 23-3 (1983), 599–616. Zbl0544.35063MR721386
  5. [5] P. D’Ancona, T. Kinoshita and S. Spagnolo, Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes, J. Math. Kyoto. Univ. 12 (1983), 434–460. Zbl0544.35063
  6. [6] P. D’Ancona, T. Kinoshita and S. Spagnolo, The Cauchy Problem for Nonlinear Hyperbolic Systems, Bull. Sci. Math. 110 (1986), 3–48. Zbl0657.35082
  7. [7] P. D’Ancona, T. Kinoshita and S. Spagnolo, Wellposedness in Gevrey class of the Cauchy problem for hyperbolic operators, Bull. Sc. Math. 111 (1987), 415–438. Zbl0653.35051
  8. [8] T. Nishitani, Sur les équations hyperboliques à coefficients hölderients en t et de classes de Gevrey en x , Bull. Sci. Math. 107 (1983), 113–138. Zbl0536.35042MR704720
  9. [9] Y. Ohya and S. Tarama, Le Problème de Cauchy à caractéristiques multiples dans la classe de Gevrey - coefficients hölderiens en t -, In: “Hyperbolic Equations and Related Topics”, Proc. Taniguchi Internat. Symp. (1984), 273–302. Zbl0665.35045
  10. [10] H. Tanabe, “Equations of Evolution”, translated from the Japanese by N. Mugibayashi and H. Haneda. Monographs and Studies in Mathematics, Vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Zbl0417.35003MR533824
  11. [11] Y. Yuzawa, The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to time, J. Differential Equations 219 (2005), 363–374. Zbl1087.35068MR2183264

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