ON CERTAIN INVERSE PROBLEMS FOR PARABOLIC EQUATIONS WITH FINAL AND INTEGRAL OBSERVATION

The inverse problems of finding the free term and the coefficient of u (x, t) in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a specia l form is proved, along with global existence, uniqueness, and stabili ty theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under re strictions in the form of inequalities containing no smallness conditi ons. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse pr oblem and the completeness of a certain system of functions is establi shed.