In This article we consider a physical model describing time-dependent heat transfer by conduction and radiation. This model contains two conducting and opaque materials which are in contact by radiation through a transparent medium bounded by diffuse-grey surfaces. The aim of this work is to present a reliable framework to prove the existence and the uniqueness of a weak solution for this problem. The existence of the solution can be proved by solving an auxiliary problem by the Galerkin-based approximation method and Moser-type arguments which implies the existence of solution to the original problem. The uniqueness of the solution will be proved by using the same approach in our previous work for the stationary heat transfer model and some ideas from nonlinear heat conduction analysis.