Nonlinear wave equations and singular solutions
Author:
Hideshi Yamane
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3659-3667
MSC (2000):
Primary 35L70, 35A20
DOI:
https://doi.org/10.1090/S0002-9939-07-08926-5
Published electronically:
August 14, 2007
MathSciNet review:
2336582
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface, which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.
- 1. Raymond Gérard and Hidetoshi Tahara, Solutions holomorphes et singulières d’équations aux dérivées partielles singulières non linéaires, Publ. Res. Inst. Math. Sci. 29 (1993), no. 1, 121–151 (French). MR 1208031, https://doi.org/10.2977/prims/1195167545
- 2. Raymond Gérard and Hidetoshi Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1757086
- 3. Takao Kobayashi, Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations, Publ. Res. Inst. Math. Sci. 34 (1998), no. 1, 43–63. MR 1617617, https://doi.org/10.2977/prims/1195144827
- 4. Satyanad Kichenassamy and Walter Littman, Blow-up surfaces for nonlinear wave equations. I, Comm. Partial Differential Equations 18 (1993), no. 3-4, 431–452. MR 1214867, https://doi.org/10.1080/03605309308820936
- 5. Satyanad Kichenassamy and Walter Littman, Blow-up surfaces for nonlinear wave equations. II, Comm. Partial Differential Equations 18 (1993), no. 11, 1869–1899. MR 1243529, https://doi.org/10.1080/03605309308820997
- 6. Satyanad Kichenassamy and Gopala Krishna Srinivasan, The structure of WTC expansions and applications, J. Phys. A 28 (1995), no. 7, 1977–2004. MR 1336507
- 7. H. Tahara, On the singularities of solutions of nonlinear partial differential equations in the complex domain, Microlocal analysis and complex Fourier analysis, World Sci. Publ., River Edge, NJ, 2002, pp. 273–283. MR 2068543, https://doi.org/10.1142/9789812776594_0019
- 8. Hidetoshi Tahara, On the singularities of solutions of nonlinear partial differential equations in the complex domain. II, Differential equations & asymptotic theory in mathematical physics, Ser. Anal., vol. 2, World Sci. Publ., Hackensack, NJ, 2004, pp. 343–354. MR 2161979, https://doi.org/10.1142/9789812702395_0010
- 9. Keisuke Uchikoshi, Singular Cauchy problems for quasilinear equations of order two, J. Math. Pures Appl. (9) 83 (2004), no. 9, 1151–1178 (English, with English and French summaries). MR 2091958, https://doi.org/10.1016/j.matpur.2004.02.009
- 10. Hideshi Yamane, Nonlinear Cauchy problems with small analytic data, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3353–3361. MR 2231920, https://doi.org/10.1090/S0002-9939-06-08410-3
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Additional Information
Hideshi Yamane
Affiliation:
Department of Physics, Kwansei Gakuin University, Gakuen 2-1, Sanda, Hyougo 669-1337, Japan
Email:
yamane@ksc.kwansei.ac.jp; yamanehideshi@95.alumni.u-tokyo.ac.jp
DOI:
https://doi.org/10.1090/S0002-9939-07-08926-5
Keywords:
Nonlinear wave equation,
nonlinear Fuchsian equation,
singular Cauchy problem
Received by editor(s):
March 21, 2006
Received by editor(s) in revised form:
September 8, 2006
Published electronically:
August 14, 2007
Additional Notes:
This research was partially supported by Grant-in-Aid for Scientific Research (No.17540182), Japan Society for the Promotion of Science.
Communicated by:
David S. Tartakoff
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
