In this paper the support of a Radon measure is selected in an optimal way. The solution of the parabolic equation depends on the mesure via the mixed type boundary conditions. The existence of a solution for a class of domain optimization problems is shown. We also investigate the behaviour of the optimal solution for some time $T$, when $T\to\infty$ and we prove that it converges to the optimal solution of the stationary problem. The first order necessary optimality conditions are derived.