On the control problem associated with the heating process in the bounded domain

The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton's law. The control parameter is equal to the magnitude of output of hot air and is defined on a givenmpart of the boundary. Then we determined the dependence $T(\theta)$ on the parameters of the temperature process when $\theta$ is close to critical value.