  The aim of this paper is to study the $d$-dimensional stochastic heat
equation with a multiplicative Gaussian noise which is white in space and it
has the covariance of a fractional Brownian motion with Hurst parameter $% H\in
(0,1)$ in time. Two types of equations are considered. First we consider the
equation in the It\^{o}-Skorohod sense, and later in the Stratonovich sense. An
explicit chaos development for the solution is obtained. On the other hand, the
moments of the solution are expressed in terms of the exponential moments of
some weighted intersection local time of the Brownian motion.
