  In this note, we establish optimal lower and upper Gaussian bounds for the
density of the solution to a class of stochastic integral equations driven by
an additive spatially homogeneous Gaussian random field. The proof is based on
the techniques of the Malliavin calculus and a density formula obtained by
Nourdin and Viens. Then, the main result is applied to the mild solution of a
general class of SPDEs driven by a Gaussian noise which is white in time and
has a spatially homogeneous correlation. In particular, this covers the case of
the stochastic heat and wave equations in $\mathbb{R}^d$ with $d\geq 1$ and
$d\leq 3$, respectively. The upper and lower Gaussian bounds have the same form
and are given in terms of the variance of the stochastic integral term in the
mild form of the equation.
