Blow up of the Solutions of Nonlinear Wave Equation
We construct for every fixed n≥2 the metric gs=h1(r)dt2−h2(r)dr2−k1(É)dÉ12−⋯−kn−1(É)dÉn−12, where h1(r), h2(r), ki(É), 1≤i≤n−1, are continuous functions, r=|x|, for which we consider the Cauchy problem (utt−Δu)gs=f(u)+g(|x|),...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2007-08-01
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| Series: | Boundary Value Problems |
| Online Access: | http://dx.doi.org/10.1155/2007/42954 |
| _version_ | 1819225042480791552 |
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| author | Svetlin Georgiev Georgiev |
| author_facet | Svetlin Georgiev Georgiev |
| author_sort | Svetlin Georgiev Georgiev |
| collection | DOAJ |
| description | We construct for every fixed n≥2 the metric gs=h1(r)dt2−h2(r)dr2−k1(É)dÉ12−⋯−kn−1(É)dÉn−12, where h1(r), h2(r), ki(É), 1≤i≤n−1, are continuous functions, r=|x|, for which we consider the Cauchy problem (utt−Δu)gs=f(u)+g(|x|), where x∈â„Ân, n≥2; u(1,x)=u∘(x)∈L2(â„Ân), ut(1,x)=u1(x)∈H˙−1(â„Ân), where f∈ðÂ’ž1(â„Â1), f(0)=0, a|u|≤f′(u)≤b|u|, g∈ðÂ’ž(â„Â+), g(r)≥0, r=|x|, a and b are positive constants. When g(r)≡0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r)=v(t)É(r) for which limt→0‖u‖L2([0,∞))=∞. When g(r)≠0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r)=v(t)É(r) for which limt→0‖u‖L2([0,∞))=∞. |
| first_indexed | 2024-12-23T10:03:18Z |
| format | Article |
| id | doaj.art-bbb0bf52626e483d8d762c555dfbce5f |
| institution | Directory Open Access Journal |
| issn | 1687-2762 1687-2770 |
| language | English |
| last_indexed | 2024-12-23T10:03:18Z |
| publishDate | 2007-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-bbb0bf52626e483d8d762c555dfbce5f2022-12-21T17:51:10ZengSpringerOpenBoundary Value Problems1687-27621687-27702007-08-01200710.1155/2007/42954Blow up of the Solutions of Nonlinear Wave EquationSvetlin Georgiev GeorgievWe construct for every fixed n≥2 the metric gs=h1(r)dt2−h2(r)dr2−k1(É)dÉ12−⋯−kn−1(É)dÉn−12, where h1(r), h2(r), ki(É), 1≤i≤n−1, are continuous functions, r=|x|, for which we consider the Cauchy problem (utt−Δu)gs=f(u)+g(|x|), where x∈â„Ân, n≥2; u(1,x)=u∘(x)∈L2(â„Ân), ut(1,x)=u1(x)∈H˙−1(â„Ân), where f∈ðÂ’ž1(â„Â1), f(0)=0, a|u|≤f′(u)≤b|u|, g∈ðÂ’ž(â„Â+), g(r)≥0, r=|x|, a and b are positive constants. When g(r)≡0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r)=v(t)É(r) for which limt→0‖u‖L2([0,∞))=∞. When g(r)≠0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r)=v(t)É(r) for which limt→0‖u‖L2([0,∞))=∞.http://dx.doi.org/10.1155/2007/42954 |
| spellingShingle | Svetlin Georgiev Georgiev Blow up of the Solutions of Nonlinear Wave Equation Boundary Value Problems |
| title | Blow up of the Solutions of Nonlinear Wave Equation |
| title_full | Blow up of the Solutions of Nonlinear Wave Equation |
| title_fullStr | Blow up of the Solutions of Nonlinear Wave Equation |
| title_full_unstemmed | Blow up of the Solutions of Nonlinear Wave Equation |
| title_short | Blow up of the Solutions of Nonlinear Wave Equation |
| title_sort | blow up of the solutions of nonlinear wave equation |
| url | http://dx.doi.org/10.1155/2007/42954 |
| work_keys_str_mv | AT svetlingeorgievgeorgiev blowupofthesolutionsofnonlinearwaveequation |