  The behavior of the single-species reaction process $A+A\to O$ is examined
near an impenetrable boundary, representing the flask containing the reactants.
Two types of dynamics are considered for the reactants: diffusive and ballistic
propagation. It is shown that the effect of the boundary is quite different in
both cases: diffusion-reaction leads to a density excess, whereas ballistic
annihilation exhibits a density deficit, and in both cases the effect is not
localized at the boundary but penetrates into the system. The field-theoretic
renormalization group is used to obtain the universal properties of the density
excess in two dimensions and below for the reaction-diffusion system. In one
dimension the excess decays with the same exponent as the bulk and is found by
an exact solution. In two dimensions the excess is marginally less relevant
than the bulk decay and the density profile is again found exactly for late
times from the RG-improved field theory. The results obtained for the diffusive
case are relevant for Mg$^{2+}$ or Cd$^{2+}$ doping in the TMMC crystal's
exciton coalescence process and also imply a surprising result for the dynamic
magnetization in the critical one-dimensional Ising model with a fixed spin.
For the case of ballistic reactants, a model is introduced and solved exactly
in one dimension. The density-deficit profile is obtained, as is the density of
left and right moving reactants near the impenetrable boundary.
