On the existence of periodic solutions of an hyperbolic equation in a thin domain

Russell Johnson; Mikhail Kamenskii; Paolo Nistri

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1997)

  • Volume: 8, Issue: 3, page 189-195
  • ISSN: 1120-6330

Abstract

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For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.

How to cite

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Johnson, Russell, Kamenskii, Mikhail, and Nistri, Paolo. "On the existence of periodic solutions of an hyperbolic equation in a thin domain." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 8.3 (1997): 189-195. <http://eudml.org/doc/244222>.

@article{Johnson1997,
abstract = {For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.},
author = {Johnson, Russell, Kamenskii, Mikhail, Nistri, Paolo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Hyperbolic nonlinear equations; Periodic solutions; Topological degree; non-autonomous and autonomous cases; admissible homotopies},
language = {eng},
month = {10},
number = {3},
pages = {189-195},
publisher = {Accademia Nazionale dei Lincei},
title = {On the existence of periodic solutions of an hyperbolic equation in a thin domain},
url = {http://eudml.org/doc/244222},
volume = {8},
year = {1997},
}

TY - JOUR
AU - Johnson, Russell
AU - Kamenskii, Mikhail
AU - Nistri, Paolo
TI - On the existence of periodic solutions of an hyperbolic equation in a thin domain
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1997/10//
PB - Accademia Nazionale dei Lincei
VL - 8
IS - 3
SP - 189
EP - 195
AB - For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.
LA - eng
KW - Hyperbolic nonlinear equations; Periodic solutions; Topological degree; non-autonomous and autonomous cases; admissible homotopies
UR - http://eudml.org/doc/244222
ER -

References

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  1. CIUPERCA, I., Lower semicontinuity of attractors for a reaction-diffusion equation on thin domains with varying order of thinness. Preprint, Université de Paris-Sud, 1996. MR1383978DOI10.1006/jdeq.1996.0051
  2. CIUPERCA, I., Reaction-diffusion equations on thin domains with varying order of thinness. Jour. Diff. Eqns., 126, 1996, 244-291. Zbl0849.35049MR1383978DOI10.1006/jdeq.1996.0051
  3. GOUROVA, I. N. - KAMENSKII, M. I., On the method of semidiscretization in periodic solutions problems for quasilinear autonomous parabolic equations. Differential Equations, 32, 1996, 101-106 (in Russian). Zbl0877.35057MR1432954
  4. HALE, J. - RAUGEL, G., A damped hyperbolic equation on thin domains. Trans. Am. Math. Soc., 329, 1992, 185-219. Zbl0761.35052MR1040261DOI10.2307/2154084
  5. HALE, J. - RAUGEL, G., Reaction-diffusion equations in thin domains. Jour. Math. Pures et Appl., 71, 1992, 33-95. Zbl0840.35044MR1151557
  6. HALE, J. - RAUGEL, G., Limits of semigroups depending on parameters. Resenhas IME-USP, 1, 1993, 1-45. Zbl0863.58046MR1257603
  7. JOHNSON, R. - KAMENSKII, M. I. - NISTRI, P., Existence of periodic solutions for an autonomous damped wave equation in a thin domain. Submitted. Zbl0910.35009
  8. JOHNSON, R. - KAMENSKII, M. I. - NISTRI, P., On periodic solutions of a damped wave equation in a thin domain using degree theoretic methods. Jour. Diff. Eqns., to appear. Zbl0890.35078MR1473860DOI10.1006/jdeq.1997.3301
  9. KRASNOSELSKII, M. - ZABREIKO, P. - PUSTYL'NIK, E. - SOBOLEVSKI, P., Integral Operators in Spaces of Summable Functions. Noordhooff International Publishing, Leyden1976. Zbl0145.39703MR385645
  10. KREIN, S., Linear Differential Equations in Banach Spaces. Nauka, Moscow1967. MR247239
  11. RAUGEL, G., Dynamics of partial differential equations in thin domains. Lecture Notes in Mathematics, Springer-Verlag, Berlin1995, 1609, 208-315. Zbl0851.58038MR1374110DOI10.1007/BFb0095241

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