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 special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. 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 problem and the completeness of a certain system of functions is established.