Degenerate hyperbolic equations with lower degree degeneracy
HTML articles powered by AMS MathViewer
- by Qing Han and Yannan Liu
- Proc. Amer. Math. Soc. 143 (2015), 567-580
- DOI: https://doi.org/10.1090/S0002-9939-2014-12288-X
- Published electronically: October 30, 2014
- PDF | Request permission
Abstract:
We prove that the Cauchy problem of degenerate hyperbolic equations is well-posed if leading coefficients are degenerate at a low degree.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- 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
- 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, DOI 10.1007/BF02392730
- 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
- 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, DOI 10.1081/PDE-100002378
- Qing Han, Energy estimates for a class of degenerate hyperbolic equations, Math. Ann. 347 (2010), no. 2, 339–364. MR 2606940, DOI 10.1007/s00208-009-0437-2
- 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, DOI 10.1090/S0002-9947-05-03791-8
- B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- Kunihiko Kajitani and Tatsuo Nishitani, The hyperbolic Cauchy problem, Lecture Notes in Mathematics, vol. 1505, Springer-Verlag, Berlin, 1991. MR 1166190, DOI 10.1007/BFb0090882
- Tatsuo Nishitani, The Cauchy problem for weakly hyperbolic equations of second order, Comm. Partial Differential Equations 5 (1980), no. 12, 1273–1296. MR 593968, DOI 10.1080/03605308008820169
- S. Tarama, On the lemma of Colombini, Jannelli and Spagnolo, Memoirs of the Faculty of Engineering, Osaka City University 41 (2000), 111–115.
Bibliographic 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
- MR Author ID: 797104
- Email: liuyn@th.btbu.edu.cn, Yannan.Liu.148@nd.edu
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3283645