  This paper attempts to obtain necessary and sufficient conditions to solve
the parabolic Anderson model with fractional Gaussian noises:
$\frac{\partial}{\partial t}u(t,x)=\frac{1}{2}\Delta
u(t,x)+u(t,x)\dot{W}(t,x)$, where $ {W}(t,x)$ is the fractional Brownian field
with temporal Hurst parameter $H_0\in [1/2, 1) $ and spatial Hurst parameters
$H$ $ =(H_1, \cdots, H_d)$ $ \in (0, 1)^d$, and $\dot{W}(t,x)=\frac{\partial
^{d+1}}{\partial t \partial x_1 \cdots \partial x_d}W(t,x)$. When $d=1$ and
when $(H_0,H)\in(\frac 12,1)\times(\frac 1{20},\frac 12)$ we show that the
condition $2H_0+H>5/2$ is necessary and sufficient to ensure the existence of a
unique solution for the parabolic Anderson Model. When $d\ge 2$, we find the
necessary and sufficient condition on the Hurst parameters so that each chaos
of the solution candidate is square integrable.
