On the Cauchy problem of degenerate hyperbolic equations

Authors:
Qing Han, Jia-Xing Hong and Chang-Shou Lin

Journal:
Trans. Amer. Math. Soc. **358** (2006), 4021-4044

MSC (2000):
Primary 35L15, 35L80

Published electronically:
September 22, 2005

MathSciNet review:
2219008

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

**1.**Ferruccio Colombini, Haruhisa Ishida, and Nicola Orrú,*On the Cauchy problem for finitely degenerate hyperbolic equations of second order*, Ark. Mat.**38**(2000), no. 2, 223–230. MR**1785400**, 10.1007/BF02384318**2.**Ferruccio Colombini and Sergio Spagnolo,*An example of a weakly hyperbolic Cauchy problem not well posed in 𝐶^{∞}*, Acta Math.**148**(1982), 243–253. MR**666112**, 10.1007/BF02392730**3.**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**, 10.1081/PDE-100002378**4.**K. O. Friedrichs,*Symmetric hyperbolic linear differential equations*, Comm. Pure Appl. Math.**7**(1954), 345–392. MR**0062932****5.**K. O. Friedrichs,*Symmetric positive linear differential equations*, Comm. Pure Appl. Math.**11**(1958), 333–418. MR**0100718****6.**Qing Han, Jia-Xing Hong, and Chang-Shou Lin,*Local isometric embedding of surfaces with nonpositive Gaussian curvature*, J. Differential Geom.**63**(2003), no. 3, 475–520. MR**2015470****7.**Lars Hörmander,*Lectures on nonlinear hyperbolic differential equations*, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR**1466700****8.**Haruhisa Ishida and Hideto Odai,*The initial value problem for some degenerate hyperbolic equations of second order in Gevrey classes*, Funkcial. Ekvac.**43**(2000), no. 1, 71–85. MR**1774369****9.**Haruhisa Ishida and Karen Yagdjian,*On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity*, Publ. Res. Inst. Math. Sci.**38**(2002), no. 2, 265–287. MR**1903740****10.**Kunihiko Kajitani and Tatsuo Nishitani,*The hyperbolic Cauchy problem*, Lecture Notes in Mathematics, vol. 1505, Springer-Verlag, Berlin, 1991. MR**1166190****11.**Karen Yagdjian,*The Cauchy problem for hyperbolic operators*, Mathematical Topics, vol. 12, Akademie Verlag, Berlin, 1997. Multiple characteristics; Micro-local approach. MR**1469977**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35L15,
35L80

Retrieve articles in all journals with MSC (2000): 35L15, 35L80

Additional Information

**Qing Han**

Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556 – and – Max-Planck Institute for Mathematics, Inselstr. 22 - 26, 04103 Leipzig, Germany

Email:
qhan@nd.edu, qinghan@mis.mpg.de

**Jia-Xing Hong**

Affiliation:
Institute of Mathematics, Fudan University, Shanghai, People’s Republic of China

Email:
jxhong@fudan.ac.cn

**Chang-Shou Lin**

Affiliation:
Department of Mathematics, National Chung-Cheng University, Ming-Hsiung, Chiayi, Taiwan

Email:
cslin@math.ccu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9947-05-03791-8

Keywords:
Degenerate hyperbolic equations,
Cauchy problems

Received by editor(s):
April 16, 2003

Received by editor(s) in revised form:
June 21, 2004

Published electronically:
September 22, 2005

Additional Notes:
The first author was supported in part by an NSF grant and a Sloan research fellowship

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.