Nonlinear wave equations and singular solutions

Yamane, Hideshi

doi:10.1090/S0002-9939-07-08926-5

Nonlinear wave equations and singular solutions
HTML articles powered by AMS MathViewer

by Hideshi Yamane PDF

Proc. Amer. Math. Soc. 135 (2007), 3659-3667 Request permission

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.

References

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, DOI 10.2977/prims/1195167545
Raymond Gérard and Hidetoshi Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1757086, DOI 10.1007/978-3-322-80284-2
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, DOI 10.2977/prims/1195144827
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, DOI 10.1080/03605309308820936
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, DOI 10.1080/03605309308820997
Satyanad Kichenassamy and Gopala Krishna Srinivasan, The structure of WTC expansions and applications, J. Phys. A 28 (1995), no. 7, 1977–2004. MR 1336507
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, DOI 10.1142/9789812776594_{0}019
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, DOI 10.1142/9789812702395_{0}010
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, DOI 10.1016/j.matpur.2004.02.009
Hideshi Yamane, Nonlinear Cauchy problems with small analytic data, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3353–3361. MR 2231920, DOI 10.1090/S0002-9939-06-08410-3