The duality principle and the uniqueness problem for some classes of parabolic equations with unknown coefficients

The conditions under which solutions to inverse problems in certain classes of smooth functions are unique with unknown coefficients and duality principles are presented. The inverse problems with final observation for linear and quasilinear parabolic equations with unknown coefficients multiplying lower-order terms or derivatives are studied. A characteristic feature of these problems is the possible nonexistence of solutions and the instability of solutions with respect to errors in input data. This approach ensures the possibility of taking into account the dependence of all coefficients of a parabolic equation on a variable in linear case and in quasilinear case. The density properties of the dual problem ensure the uniqueness of a solution of the initial inverse problem. All results presented can be generalized to boundary conditions of the third kind and to the multidimensional case.