A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R²
In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂[subscript t][superscript 2]u- (∆ℍ[superscript n] +ρ[superscript 2] )u = -|u| p[superscript -1] u, (x,t) ∈ ℍ n × ℝ, and we introduce a Morawetz-type inequality (Formula presented) where ε is the energ...
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| Format: | Article |
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American Mathematical Society (AMS)
2018
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| Online Access: | http://hdl.handle.net/1721.1/116024 https://orcid.org/0000-0002-8220-4466 |
| _version_ | 1826216914807947264 |
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| author | Shen, Ruipeng Staffilani, Gigliola |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Shen, Ruipeng Staffilani, Gigliola |
| author_sort | Shen, Ruipeng |
| collection | MIT |
| description | In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂[subscript t][superscript 2]u- (∆ℍ[superscript n] +ρ[superscript 2] )u = -|u| p[superscript -1] u, (x,t) ∈ ℍ n × ℝ, and we introduce a Morawetz-type inequality (Formula presented) where ε is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H[superscript 1/2,1/2] × H[superscript 1/2,−1/2](ℍ[superscript n) if 2 ≤ n ≤ 6 and 1 < p < p[subscript c] = 1+4/(n − 2). As another application we show that a solution to the quintic wave equation ∂[subscript t][superscript 2]u − Δu = −|u|[superscript 4] u on ℝ[superscript 2] scatters if its initial data are radial and satisfy the conditions |∇u[subscript 0](x)|, |u[subscript 1](x)| ≤ A(|x| + 1) [superscript −3/2−ε] , |u[subscript 0](x)| ≤ A(|x|)[superscript −1/2−ε], ε > 0. |
| first_indexed | 2024-09-23T16:55:27Z |
| format | Article |
| id | mit-1721.1/116024 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T16:55:27Z |
| publishDate | 2018 |
| publisher | American Mathematical Society (AMS) |
| record_format | dspace |
| spelling | mit-1721.1/1160242022-10-19T04:00:33Z A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² Shen, Ruipeng Staffilani, Gigliola Massachusetts Institute of Technology. Department of Mathematics Shen, Ruipeng Staffilani, Gigliola In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂[subscript t][superscript 2]u- (∆ℍ[superscript n] +ρ[superscript 2] )u = -|u| p[superscript -1] u, (x,t) ∈ ℍ n × ℝ, and we introduce a Morawetz-type inequality (Formula presented) where ε is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H[superscript 1/2,1/2] × H[superscript 1/2,−1/2](ℍ[superscript n) if 2 ≤ n ≤ 6 and 1 < p < p[subscript c] = 1+4/(n − 2). As another application we show that a solution to the quintic wave equation ∂[subscript t][superscript 2]u − Δu = −|u|[superscript 4] u on ℝ[superscript 2] scatters if its initial data are radial and satisfy the conditions |∇u[subscript 0](x)|, |u[subscript 1](x)| ≤ A(|x| + 1) [superscript −3/2−ε] , |u[subscript 0](x)| ≤ A(|x|)[superscript −1/2−ε], ε > 0. 2018-05-31T17:32:07Z 2018-05-31T17:32:07Z 2015-03 2018-05-30T17:20:51Z Article http://purl.org/eprint/type/JournalArticle 0002-9947 1088-6850 http://hdl.handle.net/1721.1/116024 Shen, Ruipeng and Gigliola Staffilani. “A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R².” Transactions of the American Mathematical Society 368, 4 (March 2015): 2809–2864 © 2015 American Mathematical Society https://orcid.org/0000-0002-8220-4466 http://dx.doi.org/10.1090/S0002-9947-2015-06513-1 Transactions of the American Mathematical Society Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Mathematical Society (AMS) American Mathematical Society |
| spellingShingle | Shen, Ruipeng Staffilani, Gigliola A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title | A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title_full | A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title_fullStr | A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title_full_unstemmed | A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title_short | A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R² |
| title_sort | semi linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on r² |
| url | http://hdl.handle.net/1721.1/116024 https://orcid.org/0000-0002-8220-4466 |
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