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 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.


Introduction
Boundary value problem and parabolic equations and also related models have many applications in various fields, for example, heat transfer, biology, physics. Let us refer some works concerning the existence of parabolic equations, for example [13,14,15,16,17] and the references therein.
In this paper, we consider a final value problem (often called the backward problem) for the following parabolic equation where S n is the unit sphere on R n . This backward parabolic problem is ill-posed in the sense of Hadamard. Indeed, our above problem is non well-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. If the given data is noise by the measured data with small error then the corresponding solutions may have big errors. This is disadvantage point for compute the numerical solution of the problem. Let us assume that if the given final observation data g is noisy by the data g which satisfies that Here, our main goal in this paper is to construct a regularization problem and prove that it is well-posed. Backward problem is also ill-posed and there are many publications about regularization, for example, N.H. Tuan et al. [10,11,12]. Partial Differential Equations (PDEs) on the sphere has many applications in various fields, for example, physical geodesy, geophysics, oceanography, and biology. Let us refer the reader to many papers of Q.T. Le Gia and his group [8,9]. However, to the best of our knowledge, there are limited results on backward problem of PDEs on the sphere. To the best of author's knowledge, there are very few papers on backward problem on the sphere. In order to study the models on the sphere, we can apply some techniques and knowledge on Spherical harmonics. In this paper, we apply two various methods for approximate the backward problem. In the linear case F = F (x, t), we use the Fourier truncation method. In the nonlinear case F = F (u), we use the method of quasi-reversibility.
The paper is organized as follows. In the section 3, we use truncation method to give approximate solution. In section 4, we present a quasi-reversibility regularization method and establish the convergence estimates between the regularized solution and the exact solution.

Preliminaries
From [8], we know that the eigenvalues for −∆ are as follows λ l = l(l + n − 1), l = 0, 1, 2, ...... and the eigenfunctions Y l (x) such that Let us denote the space V l which contain all spherical harmonics in the following For any function f ∈ L 2 (S n ), we have the expansion of spherical harmonics as follows Here dS is called by the surface measure of S n . The Sobolev space H σ (S n ) for σ > 0 is the space which consists of all function f such that

The inverse problem
Let u(x, T ) = g(x) and the source term F (x, t) be given. The linear backward parabolic problem is of finding u(x, 0) from the system Theorem 3.1. The Problem (3) has a unique solution if and only if the following holds Then, its solution has the form . Then u is given by where This implies that Hence Suppose that (4) holds. Define the function We consider the problem of finding a solution u from the original value u( By Theorem 1, Problem (23) has a unique solution u ∈ L 2 (S n ). It is given by Since We get So, we deduce that u is a solution of Problem (23). And we also have (5).

Truncation regularization method
In this subsection, we give a regularized solution as follows where M( ) is chosen later.
Theorem 3.2. Let g , F be as follows Let us choose M( ) = 1 m log(1/ ) for any 0 < m < 1, then we obtain Proof. Set the following function Let us first obtain the following estimate Noting that if λ l > M( ) then we get This latter inequality implies that Then, we get the following estimate Since the fact that Combining (17) and (19), we conclude that By choose M( ) = 1 m log(1/ ) and noting that 0 < m < 1, we find that which allows us to get the desired result.

Nonlinear backward parabolic on the sphere
To more clear, we discuss some details on the direct problem for nonlinear parabolic equation on the sphere.

The direct problem
The direct problem is of finding u(x, t) from the known data u(x, 0) And u has the form where S(t)f = e −t∆ * are defined as

The backward in time problem
In this section, we are looking for solution u of the following backward in time problem  Proof. Let u N and v N be two solution of problem (24) such that u N , v N ∈ C([0, T ]; H σ (S n )). Put By direct computation, we have w N k satisfying the equation It follows that Recall the the Lipchitz property of F given in Theorem 1, we have and we have Combining (26), (27), (28), we get By taking the integration with respect to s from t to T , we have This follows that Choosing k = K + λ N and noting that u This ends the proof.
Theorem 4.3. Assume that the Problem with g ∈ H σ has a weak solution u ∈ C([0, T ]; H σ (S n )). For any > 0, let g ∈ H σ (S n ) such that Suppose that F satisfies the Lipschitz condition on H σ , i.e., there exists a constant K such that for any u, v ∈ H σ . Denote by u the solution of Problem (24) with g = g .