  This paper provides necessary as well as sufficient conditions on the Hurst
parameters so that the continuous time parabolic Anderson model $\frac{\partial
u}{\partial t}=\frac{1}{2}\frac{\partial^2 u}{\partial x^2}+u\dot{W}$ on $[0,
\infty)\times {\bf R}^d $ with $d\geq 1$ has a unique random field solution,
where $W(t, x)$ is a fractional Brownian sheet on $[0, \infty)\times {\bf R}^d$
and formally $\dot W =\frac{\partial^{d+1}}{\partial t \partial x_1 \cdots
\partial x_d} W(t, x)$. When the noise $W(t, x)$ is white in time, our
condition is both necessary and sufficient when the initial data $u(0, x)$ is
bounded between two positive constants. When the noise is fractional in time
with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the
known results in literature, is different from the necessary one.
