On a heat and mass transfer model for the locally inhomogeneous initial data

We consider a model case of the problem of heat diffusion in a homogeneous body with a special initial state. The peculiarity of this initial state is its local inhomogeneity. That is, there is a closed domain $\Omega$ inside a body, the initial state is constant out of the domain. Mathematical modelling leads to the problem for a homogeneous multi-dimensional diffusion equation. We construct the boundary conditions on the boundary of the domain $\Omega$, which can be characterized as "transparent" boundary conditions. We separately consider a special case — a model of redistribution of heat in a uniform linear rod, the side surface of which is insulated in the absence of (internal and external) sources of heat and of locally inhomogeneous initial state.