  The aim of this expository paper is twofold. We start with a concise overview
of the theory of invariant slow manifolds for fast-slow dynamical systems
starting with the work by Tikhonov and Fenichel to the most recent works on
infinite-dimensional fast-slow systems. The main part focuses on a class of
linear fast-reaction PDE, which are particular forms of fast-reaction systems.
The first result shows the convergence of solutions of the linear system to the
limit system as the time-scale parameter $\varepsilon$ goes to zero. Moreover,
from the explicit solutions the slow manifold is constructed and the
convergence to the critical manifold is proven. The subsequent result, then,
states a generalized version of the Fenichel-Tikhonov theorem for linear
fast-reaction systems.
