On a final value problem for parabolic equation on the sphere with linear and nonlinear source
Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model a physical quantity on large scales. In this paper, we consider a problem of finding the initial state for backward parabolic problem on the sphere. This backward parabolic problem is ill-posed in th...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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ATNAA
2020-08-01
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| Series: | Advances in the Theory of Nonlinear Analysis and its Applications |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/tr/download/article-file/1229717 |
| _version_ | 1797919767595581440 |
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| author | Nguyen Duc Phuong Tran Thanh Binh Nguyen Hoang Luc∗ |
| author_facet | Nguyen Duc Phuong Tran Thanh Binh Nguyen Hoang Luc∗ |
| author_sort | Nguyen Duc Phuong |
| collection | DOAJ |
| description | Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model a
physical quantity on large scales. In this paper, we consider a problem of finding the initial state for
backward parabolic problem on the sphere. This backward parabolic problem is ill-posed in the sense of
Hadamard. The solutions may be not exists and if they exists then the solution does not continuous depends
on the given observation. The backward problem for homogeneous parabolic problem was recently considered
in the paper Q.T. L. Gia, N.H. Tuan, T. Tran. However, there are very few results on the backward problem
of nonlinear parabolic equation on the sphere. In this paper, we do not consider the its existence, we only
study the stability of the solution if it exists. By applying some regularized method and some techniques on
the spherical harmonics, we approximate the problem and then obtain the convalescence rate between the
regularized solution and the exact solution |
| first_indexed | 2024-04-10T13:50:46Z |
| format | Article |
| id | doaj.art-3305300730a34ebea322652702351930 |
| institution | Directory Open Access Journal |
| issn | 2587-2648 2587-2648 |
| language | English |
| last_indexed | 2024-04-10T13:50:46Z |
| publishDate | 2020-08-01 |
| publisher | ATNAA |
| record_format | Article |
| series | Advances in the Theory of Nonlinear Analysis and its Applications |
| spelling | doaj.art-3305300730a34ebea3226527023519302023-02-15T16:10:41ZengATNAAAdvances in the Theory of Nonlinear Analysis and its Applications2587-26482587-26482020-08-014314315110.31197/atnaa.753458On a final value problem for parabolic equation on the sphere with linear and nonlinear sourceNguyen Duc PhuongTran Thanh BinhNguyen Hoang Luc∗Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model a physical quantity on large scales. In this paper, we consider a problem of finding the initial state for backward parabolic problem on the sphere. This backward parabolic problem is ill-posed in the sense of Hadamard. The solutions may be not exists and if they exists then the solution does not continuous depends on the given observation. The backward problem for homogeneous parabolic problem was recently considered in the paper Q.T. L. Gia, N.H. Tuan, T. Tran. However, there are very few results on the backward problem of nonlinear parabolic equation on the sphere. In this paper, we do not consider the its existence, we only study the stability of the solution if it exists. By applying some regularized method and some techniques on the spherical harmonics, we approximate the problem and then obtain the convalescence rate between the regularized solution and the exact solutionhttps://dergipark.org.tr/tr/download/article-file/1229717cauchy problemparabolic on the sphereill-posed problemconvergence estimates |
| spellingShingle | Nguyen Duc Phuong Tran Thanh Binh Nguyen Hoang Luc∗ On a final value problem for parabolic equation on the sphere with linear and nonlinear source Advances in the Theory of Nonlinear Analysis and its Applications cauchy problem parabolic on the sphere ill-posed problem convergence estimates |
| title | On a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| title_full | On a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| title_fullStr | On a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| title_full_unstemmed | On a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| title_short | On a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| title_sort | on a final value problem for parabolic equation on the sphere with linear and nonlinear source |
| topic | cauchy problem parabolic on the sphere ill-posed problem convergence estimates |
| url | https://dergipark.org.tr/tr/download/article-file/1229717 |
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