On a final value problem for parabolic equation on the sphere with linear and nonlinear source

Yıl 2020, Cilt: 4 Sayı: 3, 143 - 151, 31.08.2020

Nguyen Duc Phuong Tran Binh Nguyen Luc

https://doi.org/10.31197/atnaa.753458

Cited By: 1

Öz

 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.

Anahtar Kelimeler

Cauchy problem, parabolic on the sphere, Ill-posed problem, Convergence estimates

Kaynakça

• \bibitem {Tuan} N.H. Tuan, L.D. Long, S. Tatar, \emph { Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation} Appl. Anal. 97 (2018), no. 5, 842--863
• \bibitem {Gia} Q.T. Le Gia \emph{Approximation of parabolic PDEs on spheres using collocation method,} Adv. Comput. Math., 22 (2005), 377--397.
• \bibitem {Tuan} D.D. Trong, N.H. Tuan, \emph{ A nonhomogeneous backward heat problem: Regularization and error estimates,} Vol. 2008(2008), No. 33, pp. 1--14.
• \bibitem{Tuan1} D.D. Trong, N.H. Tuan, P.H. Quan, \emph{A quasi-boundary value method for regularizing nonlinear ill-posed problems,} Vol. 2009(2009), No. 109, pp. 1--16.
• \bibitem {Sa} K. Sakamoto, M. Yamamoto, \emph{ Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems} J. Math. Anal. Appl. 382 (2011), no. 1, 426--447
• \bibitem {Clean} W. McLean, \emph{Regularity of solutions to a time-fractional diffusion equation} ANZIAM J. 52 (2010), no. 2, 123--138.
• \bibitem {Tuan3} Q.T. L. Gia, N.H. Tuan, T. Tran, \emph{Solving the backward heat equation on the unit sphere} ANZIAM J. (E) 56 (2016), pp. C262--C278.
• \bibitem {Thong} Q.T. L. Gia,\emph{ Approximation of parabolic PDEs on spheres using collocation method,} Adv. Comput. Math., 22 (2005), 377-397.
• \bibitem {Thong1} Q.T. L. Gia,\emph{ Galerkin approximation of elliptic PDEs on spheres} J. Approx. Theory , 130 (2004), 125-149
• \bibitem {Tuan4} N.H. Tuan, D.D. Trong, \emph{ On a backward parabolic problem with local Lipschitz source} J. Math. Anal. Appl. 414 (2014), no. 2, 678--692.
• \bibitem {Tuan5} N.H. Tuan, V.V. Au, V.A. Khoa, D. Lesnic, \emph{ Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction} Inverse Problems 33 (2017), no. 5, 055019, 40 pp.
• \bibitem {Tuan6} N.H. Tuan, V.V. Au, V.A. Khoa, \emph{Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements} SIAM J. Math. Anal. 51 (2019), no. 1, 60--85.
• \bibitem {Geo} S. G. Georgiev, K. Zenir, \emph{New results on IBVP for Class of Nonlinear Parabolic Equations,} (2018), Advances in the Theory of Nonlinear Analysis and its Application, Volume 2 , Issue 4 , Jan 2018 , 202 -- 216
• \bibitem {Jonna} J. M. Jonnalagadda, \emph{ Existence Results for Solutions of Nabla Fractional Boundary Value Problems with General Boundary Conditions,} Advances in the Theory of Nonlinear Analysis and its Application, Pages 29 - 42
• \bibitem {Ham} K. Hamdache, D. Hamroun, \emph{Asymptotic Behaviours for the Landau-Lifshitz-Bloch Equation,} Adv. Theory Nonlinear Anal. Appl. 3 (2019), 174–-191.