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

Authors : Nguyen Duc PHUONG, Tran BİNH, Nguyen LUC

Pages : 143-151

Doi:10.31197/atnaa.753458

View : 9 | Download : 11

Publication Date : 2020-08-31

Article Type : Research Paper

Abstract :  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.
Keywords : Cauchy problem, parabolic on the sphere, Ill posed problem, Convergence estimates