  In this paper, we observe how the heat equation in a non-cylindrical domain
can arise as the asymptotic limit of a parabolic problem in a cylindrical
domain, by adding a potential that vanishes outside the limit domain. This can
be seen as a parabolic version of a previous work by the first and last
authors, concerning the stationary case. We provide a strong convergence result
for the solution by use of energetic methods and $\Gamma$-convergence technics.
Then, we establish an exponential decay estimate coming from an adaptation of
an argument due to B. Simon.
