On the asymptotic behavior of inverse problems for parabolic equation

We study two inverse initial-boundary value problems for a linear parabolic equation. These equations arise in mathematical modeling of the viscous heat-conducting fluid motion with two or one free boundaries. The unknown function of time enters the right-hand side of the equation additively and is found from the additional condition of integral overdetermination. For both problems, a priori estimates of solutions in the uniform metric are obtained. Stationary solutions are found. Sufficient conditions for the input data, under which the solutions with increasing time tend to the stationary regime according to the exponential law, are established.