Stability of the Hartree equation with time-dependent coefficients
Abstract In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients i ∂ t u + Δ u + α ( t ) 1 | x | u + β ( t ) ( W ∗ | u | 2 ) u = 0 . $$i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{ \vert x \vert }u+\beta(t) \bigl(W \ast \vert u \vert ^{2}\bigr)u=...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-017-0854-1 |
| _version_ | 1819046941507452928 |
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| author | Binhua Feng Honghong Zhang Yanjun Zhao |
| author_facet | Binhua Feng Honghong Zhang Yanjun Zhao |
| author_sort | Binhua Feng |
| collection | DOAJ |
| description | Abstract In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients i ∂ t u + Δ u + α ( t ) 1 | x | u + β ( t ) ( W ∗ | u | 2 ) u = 0 . $$i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{ \vert x \vert }u+\beta(t) \bigl(W \ast \vert u \vert ^{2}\bigr)u=0. $$ We first obtain the Lipschitz continuity of the solution u = u ( α , β ) $u=u(\alpha ,\beta)$ with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients. Our results improve some recent results. |
| first_indexed | 2024-12-21T10:52:28Z |
| format | Article |
| id | doaj.art-3446db37932d439eb596e3fa4602ac16 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-21T10:52:28Z |
| publishDate | 2017-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-3446db37932d439eb596e3fa4602ac162022-12-21T19:06:37ZengSpringerOpenBoundary Value Problems1687-27702017-08-01201711910.1186/s13661-017-0854-1Stability of the Hartree equation with time-dependent coefficientsBinhua Feng0Honghong Zhang1Yanjun Zhao2Department of Mathematics, Northwest Normal UniversityDepartment of Mathematics, Northwest Normal UniversityCollege of Humanities and Sciences, Northeast Normal UniversityAbstract In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients i ∂ t u + Δ u + α ( t ) 1 | x | u + β ( t ) ( W ∗ | u | 2 ) u = 0 . $$i\partial_{t}u+\Delta u+ \alpha(t)\frac{1}{ \vert x \vert }u+\beta(t) \bigl(W \ast \vert u \vert ^{2}\bigr)u=0. $$ We first obtain the Lipschitz continuity of the solution u = u ( α , β ) $u=u(\alpha ,\beta)$ with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients. Our results improve some recent results.http://link.springer.com/article/10.1186/s13661-017-0854-1nonlinear Hartree equationstabilitytime-dependent coefficientsLipschitz continuity |
| spellingShingle | Binhua Feng Honghong Zhang Yanjun Zhao Stability of the Hartree equation with time-dependent coefficients Boundary Value Problems nonlinear Hartree equation stability time-dependent coefficients Lipschitz continuity |
| title | Stability of the Hartree equation with time-dependent coefficients |
| title_full | Stability of the Hartree equation with time-dependent coefficients |
| title_fullStr | Stability of the Hartree equation with time-dependent coefficients |
| title_full_unstemmed | Stability of the Hartree equation with time-dependent coefficients |
| title_short | Stability of the Hartree equation with time-dependent coefficients |
| title_sort | stability of the hartree equation with time dependent coefficients |
| topic | nonlinear Hartree equation stability time-dependent coefficients Lipschitz continuity |
| url | http://link.springer.com/article/10.1186/s13661-017-0854-1 |
| work_keys_str_mv | AT binhuafeng stabilityofthehartreeequationwithtimedependentcoefficients AT honghongzhang stabilityofthehartreeequationwithtimedependentcoefficients AT yanjunzhao stabilityofthehartreeequationwithtimedependentcoefficients |