Carleman Estimates and Inverse Problems for Non linear Partial Differential Equations

Abstract

newline The research reported in this thesis deals with two types of stability results concerning the inverse problem for non-linear parabolic systems as well as nonlinear higher-order KdV equations. More precisely, we establish a Carleman type estimate for the given problem in terms of the boundary observation and internal observation with Dirichlet and Dirichlet-Neumann types of boundary conditions. First, we establish the internal type stability results of an inverse problem of determining the damping coefficient in the Kawahara equation with the aid of internal type Carleman estimate along with certain regularity results. We then address the inverse problem of retrieving a space-dependent source term in the seventh-order generalized Korteweg de Vries equation by deriving a new Carleman estimate for the seventh-order linear operator as well as the regularity results for the direct problem. Using these estimates, we study the boundary stability results for the given model. Finally, we establish the Lipschitz stability results of an inverse problem of identification of the semilinear coefficient in the Cahn-Hilliard type tumor growth model. To evaluate the nonlinear coefficient, we derive a new boundary type Carleman estimate for the tumor growth model. Furthermore, the regularity results of tumor growth model plays a major role in the proof of stability results.