Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Degenerate hyperbolic equations with lower degree degeneracy


Authors: Qing Han and Yannan Liu
Journal: Proc. Amer. Math. Soc. 143 (2015), 567-580
MSC (2010): Primary 35L15, 35L80
DOI: https://doi.org/10.1090/S0002-9939-2014-12288-X
Published electronically: October 30, 2014
MathSciNet review: 3283645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Cauchy problem of degenerate hyperbolic equations is well-posed if leading coefficients are degenerate at a low degree.


References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces.: Pure and Applied Mathematics, Vol. 65., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 (56 #9247)
  • [2] F. Colombini, E. Jannelli, and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 291-312. MR 728438 (85f:35131)
  • [3] Ferruccio Colombini and Sergio Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in $ C^{\infty }$, Acta Math. 148 (1982), 243-253. MR 666112 (83m:35085), https://doi.org/10.1007/BF02392730
  • [4] Piero D'Ancona, Well posedness in $ C^\infty $ for a weakly hyperbolic second order equation, Rend. Sem. Mat. Univ. Padova 91 (1994), 65-83. MR 1289632 (95g:35106)
  • [5] Piero D'Ancona and Paola Trebeschi, On the local solvability for a nonlinear weakly hyperbolic equation with analytic coefficients, Comm. Partial Differential Equations 26 (2001), no. 5-6, 779-811. MR 1843284 (2002d:35132), https://doi.org/10.1081/PDE-100002378
  • [6] Qing Han, Energy estimates for a class of degenerate hyperbolic equations, Math. Ann. 347 (2010), no. 2, 339-364. MR 2606940 (2011b:35345), https://doi.org/10.1007/s00208-009-0437-2
  • [7] Qing Han, Jia-Xing Hong, and Chang-Shou Lin, On the Cauchy problem of degenerate hyperbolic equations, Trans. Amer. Math. Soc. 358 (2006), no. 9, 4021-4044. MR 2219008 (2006m:35258), https://doi.org/10.1090/S0002-9947-05-03791-8
  • [8] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay, 1967. MR 0212575 (35 #3446)
  • [9] Vladimir G. Maz'ja, Sobolev spaces.: Translated from the Russian by T. O. Shaposhnikova., Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. MR 817985 (87g:46056)
  • [10] Kunihiko Kajitani and Tatsuo Nishitani, The hyperbolic Cauchy problem, Lecture Notes in Mathematics, vol. 1505, Springer-Verlag, Berlin, 1991. MR 1166190 (94f:35074)
  • [11] Tatsuo Nishitani, The Cauchy problem for weakly hyperbolic equations of second order, Comm. Partial Differential Equations 5 (1980), no. 12, 1273-1296. MR 593968 (82i:35107), https://doi.org/10.1080/03605308008820169
  • [12] S. Tarama, On the lemma of Colombini, Jannelli and Spagnolo, Memoirs of the Faculty of Engineering, Osaka City University 41 (2000), 111-115.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35L15, 35L80

Retrieve articles in all journals with MSC (2010): 35L15, 35L80


Additional Information

Qing Han
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556 – and – Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People’s Republic of China
Email: qhan@nd.edu, qhan@math.pku.edu.cn

Yannan Liu
Affiliation: Department of Mathematics, Beijing Technology and Business University, Beijing 100048, People’s Republic of China – and – Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: liuyn@th.btbu.edu.cn, Yannan.Liu.148@nd.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12288-X
Received by editor(s): October 4, 2009
Received by editor(s) in revised form: January 26, 2013
Published electronically: October 30, 2014
Additional Notes: The first author acknowledges the support of NSF Grant DMS-1105321
The second author acknowledges the support of NSFC Grant 11201011, BNSF Grant 1132002 and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (CIT&TCD201304029)
The authors would like to thank the referees for many helpful suggestions.
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society