Theoretical And Numerical Researches For The Inverse Coefficient Problems Of Second Order Degenerate Parabolic Equations

In this paper, we mainly discuss some inverse coefficient problems for second-order degenerate parabolic equations. Under some appropriate additional condi-tions, we will study the uniqueness of the solution, the existence, uniqueness, sta-bility of the solution for the corresponding optimal control problem, and stable numerical computation methods for the solution of the inverse problem. Such kinds of problems have great significance in the fields of population prediction and control, porous media fluid mechanics, financial mathematics and other applied science. Be-ing different from ordinary inverse coefficient problems in parabolic equations, there exists degeneracy on a part of boundaries in the mathematical model. On one hand, the degeneracy may cause the corresponding boundary conditions missing; on the other hand, it can also make that the solution may has no good regularity. More-over, due to the ill-posedness for inverse problems, arbitrarily small changes in the final measurement data may lead to arbitrarily large changes in the solution.In the first chapter, we introduce the mathematical models and their research background.Some lemmas, theorems and preliminary knowledge regarding to the second-order linear partial differential equation are given in the second chapter.In the third chapter, we discuss an inverse problem of identifying the radiation coefficient in a second-order degenerate parabolic equation using the final obser-vations. The uniqueness of the original problem is proved, and then the inverse problem is transformed into an optimization problem on the basis of optimal con-trol framework. The existence and necessary condition of the optimal solution are obtained. Since the control functional is non-convex, we can not guarantee the uniqueness of the optimal solution. Assuming that T is relatively small, we succeed in proving the uniqueness and stability of the minimizer by utilizing the necessary condition and some prior estimates of the direct problem.In the fourth chapter, we mainly from the numerical analysis angle discuss an inverse problem of reconstructing the first-order term coefficient in a second-order degenerate parabolic equation using the extra condition imposed in the domain. Such a problem has obvious financial background, and is of great importance in the field of financial derivatives pricing. Similarly, we first prove the uniqueness of solution for the inverse problem, which illustrate the statement of the problem is correct. Then an implicit scheme on the basis of finite difference method is designed to obtain the numerical solution of the direct problem. For the inverse problem, we apply the predictor-corrector method to construct an iterative algorithm, and propose two numerical methods for the numerical differential problem arising in the iterative procedure. Finally, the numerical experiments are also done and the corresponding numerical results show that the algorithm is stable and it converges very quickly.In the fifth chapter, we discuss the well-posedness of the minimizer of a bi-nary functional. This problem originates from an inverse problem of simultaneously reconstructing the initial value and radiation coefficient in a heat conduction equa-tion. Being different from other ordinary one-variable control problems, the cost functional constructed in the paper contains two independent unknown functions and two independent regularization parameters. We first derive the necessary con-dition which should be satisfied by the minimizer of the cost functional. Then we use the method of estimating by part to prove the uniqueness and stability of the minimizer on the condition of that the parameter T is assumed to be relatively small.