This paper investigates a known problem of gas permeation into a finite volume. The literature treatment of this problem provides a solution, but misses the well-posedness analysis, which is of crucial importance, especially when a nonstandard problem is formulated. We have demonstrated that the Hadamard sense of well-posedness for this problem is satisfied by making use of the Duhamel's theorem. The solution we have obtained using the Duhamel's theorem and the solution available in the literature have different analytical representations, yet they are mathematically equivalent. In parallel, this paper provides some practical comments on the well-posedness analysis.