  We consider heat semigroups of the form $\exp(t(\Delta -
\lambda\mathbf{1}_{\Omega_0}))$ on bounded domains. Using variants of Moser
iteration, we show sub-exponential decay in the ``large coupling limit", i.e.
as $\lambda\nearrow\infty$, in compact subdomains of the ``obstacle",
$\Omega_0$.
