Inverse Coefficient Problems For Second Order Parabolic Equations With Singular Or Degenerate Properties

This thesis mainly considers some inverse coefficients problems of second order partial differential equations(PDEs) with singular or degenerate properties. Under some appropriate additional conditions, we will investigate the uniqueness of the solution, the existence, uniqueness, stability and convergence of the solution for the corresponding regularized problem, and effective numerical reconstruction methods for the solution of the inverse problem.In Chapter 1, we firstly introduce the background of inverse coefficients problems of PDEs, and the mathematical models arising in the thesis. Then, we elaborate the motivation and main difficulties of the research.In Chapter 2, we introduce some function spaces and the corresponding integral imbedding theory, and some well-posedness results of the second order parabolic equations which are quite important in the later proofs.In Chapter 3, we consider an inverse problem of identifying the radiative coefficient in a second order parabolic equation utilizing the terminal observations.Unlike other terminal control problems, the observation data are only given for a fixed direction rather than for the whole domain, which may make the conjugate theory for parabolic equations ineffective. Moreover, since the definite domain is of circular or sectorial type, it can be transformed into a rectangle under the polar coordinates, but in the meantime the transformation may make the principle coefficient of the equation singular. To conquer the difficulty of coefficient singularity,we propose some weighted Sobolev spaces. Based on the optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the the necessary conditions that must be satisfied by the minimizer is also deduced. By using the the necessary conditions and some a-priori estimates of the solution of the forward problem, we prove the uniqueness and stability of the minimizer. Finally, in order to illustrate the difference between the solution of the optimal control problem and that of the original problem, we prove the the convergence of the minimizer, and give the convergence rate.In Chapter 4, we investigate an inverse problem of simultaneously reconstructing the initial value and source coefficient in a second order degenerate parabolic equation using some additional conditions. The problem has the following two main features:(i) the principle coefficient of the equation degenerates into zero on both extremities of the definite domain;(ii) the mathematical model contains two independent unknown functions, and thus this is a multi-parameters inversion problem. On one hand, the degeneracy may lead to the corresponding boundary conditions missing; on the other hand, it can also cause that the solution of the equation has no sufficient regularity. Firstly, by using the Carleman estimate and the logarithmic convexity method, we prove the uniqueness and conditional stability of the solution of the original problem. Due to the ill-posedness of the original problem, we transform it into an optimal control problem by the optimization method. We establish the existence, uniqueness and convergence of the regularized solution. Since the cost functional contains two independent unknown functions and their status are different, the conjugate theory cannot be applied for our problem. Otherwise, we cannot obtain the global uniqueness of the regularized solution. Here, we use the method of estimating by part. According to the careful analysis for the necessary conditions, we finally obtain the global uniqueness and stability of the regularized solution.In Chapter 5, we discuss the numerical reconstruction of the inverse problem proposed in the previous chapter. We utilize the Landweber iteration algorithm to obtain the numerical solution of the inverse problem, where the key point is to solve the specific form of the adjoint operator for the forward problem. However,due to the coupling of the two unknown functions, it is quite difficult to directly find the structure of the adjoint operator. The operator decomposition method is applied to conquer the difficulty. The forward operator is decomposed into four independent operators, and then every adjoint operator is resolved respectively. In the end, the adjoint operator for the forward problem is obtained by combining the four adjoint operators together. The numerical experiments are done, and some typical numerical examples are also presented. Numerical results show that ouralgorithm is stable and effective, and the two unknown functions are reconstructed quite well.