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Blow up of the Solutions of Nonlinear Wave Equation
Boundary Value Problems volume 2007, Article number: 042954 (2007)
Abstract
We construct for every fixed the metric, where,,,, are continuous functions,, for which we consider the Cauchy problem, where,;,, where,,,,,, and are positive constants. When, we prove that the above Cauchy problem has a nontrivial solution in the form for which. When, we prove that the above Cauchy problem has a nontrivial solution in the form for which.
References
Jörgens K: Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Mathematische Zeitschrift 1961, 77: 295–308. 10.1007/BF01180181
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.
Rauch J: The -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.
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.
John F: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Mathematica 1979,28(1–3):235–268.
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.
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Georgiev, S.G. Blow up of the Solutions of Nonlinear Wave Equation. Bound Value Probl 2007, 042954 (2007). https://doi.org/10.1155/2007/42954
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DOI: https://doi.org/10.1155/2007/42954