  We consider the stochastic heat equation with multiplicative noise
$u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where
$\diamond$ denotes the Wick product, and the solution is interpreted in the
mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq
1/2$), and colored in space (with spatial covariance kernel $f$). We prove that
if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order
$\alpha<d$, then the sufficient condition for the existence of the solution is
$d \leq 2+\alpha$ (if $H>1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas
if $f$ is the heat kernel or the Poisson kernel, then the equation has a
solution for any $d$. We give a representation of the $k$-th order moment of
the solution, in terms of an exponential moment of the "convoluted weighted"
intersection local time of $k$ independent $d$-dimensional Brownian motions.
