Open Access

Blow up of the Solutions of Nonlinear Wave Equation

Boundary Value Problems20072007:042954

DOI: 10.1155/2007/42954

Received: 14 March 2007

Accepted: 26 May 2007

Published: 14 August 2007

Abstract

We construct for every fixed https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq1_HTML.gif the metric https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq6_HTML.gif , are continuous functions, https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq7_HTML.gif , for which we consider the Cauchy problem https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq8_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq10_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq12_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq20_HTML.gif are positive constants. When https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq21_HTML.gif , we prove that the above Cauchy problem has a nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq22_HTML.gif in the form https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq23_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq24_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq25_HTML.gif , we prove that the above Cauchy problem has a nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq26_HTML.gif in the form https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq27_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2007%2F42954/MediaObjects/13661_2007_Article_647_IEq28_HTML.gif .

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Authors’ Affiliations

(1)
Department of Differential Equations, University of Sofia

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Copyright

© 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.