This paper studies the nonlinear one-dimensional stochastic heat equation
driven by a Gaussian noise which is white in time and which has the covariance
of a fractional Brownian motion with Hurst parameter
1/4\textless{}H\textless{}1/2 in the space variable. The existence and
uniqueness of the solution u are proved assuming the nonlinear coefficient is
differentiable with a Lipschitz derivative and vanishes at 0. In the case of a
multiplicative noise, that is the linear equation, we derive the Wiener chaos
expansion of the solution and a Feynman-Kac formula for the moments of the
solution. These results allow us to establish sharp lower and upper asymptotic
bounds for the moments of the solution