  In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot
B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a
Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in
(1/2,1)$, and colored in space, with spatial covariance given by a function
$f$. Our main result gives the necessary and sufficient condition on $H$ for
the existence of the process solution. When $f$ is the Riesz kernel of order
$\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of
the condition $H>d/4$ encountered when the noise $\dot B$ is white in space.
When $f$ is the Bessel kernel or the heat kernel, the condition remains
$H>d/4$.
