Boundary Value Problems

, 2007:042954 | Cite as

Blow up of the Solutions of Nonlinear Wave Equation

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Research Article
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Abstract

We construct for every fixed Open image in new window the metric Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , are continuous functions, Open image in new window , for which we consider the Cauchy problem Open image in new window , where Open image in new window , Open image in new window ; Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window are positive constants. When Open image in new window , we prove that the above Cauchy problem has a nontrivial solution Open image in new window in the form Open image in new window for which Open image in new window . When Open image in new window , we prove that the above Cauchy problem has a nontrivial solution Open image in new window in the form Open image in new window for which Open image in new window .

Keywords

Differential Equation Continuous Function Partial Differential Equation Ordinary Differential Equation Positive Constant 

References

  1. 1.
    Jörgens K: Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Mathematische Zeitschrift 1961, 77: 295–308. 10.1007/BF01180181MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Shatah J, Struwe M: Geometric Wave Equations, Courant Lecture Notes in Mathematics. Volume 2. New York University, Courant Institute of Mathematical Sciences, New York, NY, USA; 1998:viii+153.Google Scholar
  3. 3.
    Rauch J: The Open image in new window-Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations. In Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. I, Research Notes in Math. Volume 53. Edited by: Breizis H, Lions JL. Pitman, Boston, Mass, USA; 1981:335–364.Google Scholar
  4. 4.
    Georgiev SG: Blow up of solutions for Klein-Gordon equations in the Reissner-Nordström metric. Electronic Journal of Differential Equations 2005,2005(67):1–22.Google Scholar
  5. 5.
    John F: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Mathematica 1979,28(1–3):235–268.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Georgiev SG: Blow up of the solutions of nonlinear wave equation in Reissner-Nordström metric. Dynamics of Partial Differential Equations 2006,3(4):295–329.MATHMathSciNetGoogle Scholar

Copyright information

© Svetlin Georgiev Georgiev 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Differential EquationsUniversity of SofiaSofiaBulgaria

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