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

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.