Vlasov V. Coefficient inverse problems for two-dimensional degenerate parabolic equations

The thesis deals with coefficient inverse problems for two-dimensional degenerate parabolic equations. In all problems unknown major coefficients of the equation are time-dependent and degenerate in the initial moment of time. Both isotropic and anisotropic equation types are studied, where anisotropy means a presence of varying degrees of degeneration at different spatial components in the major coefficient of the equation. Cases of weak and strong degeneration are considered. An inverse problem for a weakly degenerate two-dimensional parabolic equation with Dirichlet boundary conditions is considered. Conditions for global existence and uniqueness of a solution to such problem are established. A weakly degenerate parabolic equation with minor coefficients is studied. An inverse problem for such equation with mixed Dirichlet-Neumann boundary conditions is considered. Local existence and global uniqueness of a solution are proven. An inverse problem for a an anisotropic weakly degenerate parabolic equation with minor coefficients is considered. Conditions for the solvability of such inverse problem are established in case of Dirichlet boundary conditions. Existence and uniqueness of a global solution for an inverse problem with mixed Dirichlet-Neumann boundary conditions for a strongly degenerate isotropic parabolic equation is proven. Conditions for local existence and global uniqueness of a solution to an inverse problem with mixed Dirichlet-Neumann boundary conditions and integral overdetermination conditions for a strongly degenerate anisotropic parabolic equation are established.