Journal of the Mathematical Society of Japan

Logarithmic singularities of solutions to nonlinear partial differential equations

Hidetoshi TAHARA and Hideshi YAMANE

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Abstract

We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T.Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is done by the reduction to a Fuchsian equation with singular coefficients.

Article information

Source
J. Math. Soc. Japan, Volume 60, Number 2 (2008), 603-630.

Dates
First available in Project Euclid: 30 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1212156664

Digital Object Identifier
doi:10.2969/jmsj/06020603

Mathematical Reviews number (MathSciNet)
MR2421990

Zentralblatt MATH identifier
1151.35305

Subjects
Primary: 35A20: Analytic methods, singularities
Secondary: 35L70: Nonlinear second-order hyperbolic equations

Keywords
singular solutions Fuchsian equations logarithmic singularities nonlinear wave equations

Citation

TAHARA, Hidetoshi; YAMANE, Hideshi. Logarithmic singularities of solutions to nonlinear partial differential equations. J. Math. Soc. Japan 60 (2008), no. 2, 603--630. doi:10.2969/jmsj/06020603. https://projecteuclid.org/euclid.jmsj/1212156664


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References

  • M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series 149, Cambridge University Press, 1996.
  • R. Gérard and H. Tahara, Solutions holomorphes et singulières d'équations aux dérivées partielles singulières non linéaires, Publ. Res. Inst. Math. Sci., 29 (1993), 121–151.
  • R. Gérard and H. Tahara, Singular nonlinear partial differential equations, Vieweg, 1996.
  • L. Hörmander, Linear partial differential operators, Springer-Verlag, 1963.
  • S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, I, Comm. Partial Differential Equations, 18 (1993), 431–452.
  • S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, II, Comm. Partial Differential Equations, 18 (1993), 1869–1899.
  • S. Kichenassamy and G. K. Srinivasan, The structure of WTC expansions and applications, J. Phys. A, 28 (1995), 1977–2004.
  • S. Kichenassamy and A. D. Rendall, Analytic description of singularities in Gowdy spacetimes, Classical Quantum Gravity, 15 (1998), 1339–1355.
  • T. Kobayashi, Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations, Publ. Res. Inst. Math. Sci., 34 (1998), 43–63.
  • M. Nagumo, Über das Anfangswertproblem Parteller Differentialgleichungen, Japan. J. Math., 18 (1941), 41–47.
  • 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., Hackensack, NJ, 2002, pp.,273–283.
  • H. 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., World Sci. Publ., Hackensack, NJ, 2004, pp.,343–354.
  • H. Tahara and H. Yamazawa, Structure of solutions of nonlinear partial differential equations of Gerard-Tahara type, Publ. Res. Inst. Math. Sci., 41 (2005), 339–373.
  • H. Yamane, Nonlinear wave equations and singular solutions, Proc. Amer. Math. Soc., 135 (2007), 3659–3667.