The work is connected with investigation of nonlinear problems for parabolic equations with an unknown coefficient at the derivative with respect to time. The considered statements are new subjects in the theory of parabolic equations which essentially differ from usual boundary value problems. One of the statements is a system containing a boundary value problem of the first kind and an equation for a time dependence of the sought coefficient. For such a nonlinear system we determine the faithful character of differential relations in a class of smooth functions and establish conditions of unique solvability. The obtained results are then used for investigation of another statement in which, moreover, it is required to determine a boundary function in one of the boundary conditions by using an additional information about the sought coefficient at the final time. The nonlinear parabolic problems considered in the present work are important not only as new theoretical subjects but also as the mathematical models of physical-chemical processes with changeable inner characteristics.