  In this note, we consider the parabolic Anderson model on $\mathbb{R}_{+}
\times \mathbb{R}$, driven by a Gaussian noise which is fractional in time with
index $H_0>1/2$ and fractional in space with index $0<H<1/2$ such that
$H_0+H>3/4$. Under a general condition on the initial data, we prove the
existence and uniqueness of the mild solution and obtain its exponential upper
bounds in time for all $p$-th moments with $p\ge 2$.
