TY  - JOUR
T1  - On a  final value problem for parabolic equation  on the sphere with linear and  nonlinear source
AU  - Luc, Nguyen
AU  - Phuong, Nguyen Duc
AU  - Binh, Tran
PY  - 2020
DA  - August
DO  - 10.31197/atnaa.753458
JF  - Advances in the Theory of Nonlinear Analysis and its Application
JO  - ATNAA
PB  - Erdal KARAPINAR
WT  - DergiPark
SN  - 2587-2648
SP  - 143
EP  - 151
VL  - 4
IS  - 3
LA  - en
AB  - Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model aphysical quantity on large scales. In this paper, we consider a problem of finding the initial state forbackward parabolic problem on the sphere. This backward parabolic problem is ill-posed in the sense ofHadamard. The solutions may be not exists and if they exists then the solution does not continuous dependson the given observation. The backward problem for homogeneous parabolic problem was recently consideredin the paper Q.T. L. Gia, N.H. Tuan, T. Tran. However, there are very few results on the backward problemof nonlinear parabolic equation on the sphere. In this paper, we do not consider the its existence, we onlystudy the stability of the solution if it exists. By applying some regularized method and some techniques onthe spherical harmonics, we approximate the problem and then obtain the convalescence rate between theregularized solution and the exact solution.
KW  - Cauchy problem
KW  - parabolic on the sphere
KW  - Ill-posed problem
KW  - Convergence estimates
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CR  - \bibitem {Gia}  Q.T. Le Gia \emph{Approximation of parabolic PDEs on spheres using collocation method,}  Adv. Comput. Math., 22 (2005), 377--397.
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UR  - https://doi.org/10.31197/atnaa.753458
L1  - https://dergipark.org.tr/en/download/article-file/1229717
ER  -