A Note on Moser Iteration and the Large Coupling Limit

We consider heat semigroups of the form $\exp(t(\Delta - \lambda\mathbf{1}_{\Omega_0}))$ on bounded domains. These singularly perturbed equations arise in certain models of diffusion limited chemical reactions. 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", $\Omega_0$.