  In this article, we study a class of stochastic partial differential
equations with fractional differential operators subject to some
time-independent multiplicative Gaussian noise. We derive sharp conditions,
under which a unique global $L^p(\Omega)$-solution exits for all $p\ge 2$. In
this case, we derive exact moment asymptotics following the same strategy in a
recent work by Balan et al [1]. In the case when there exits only a local
solution, we determine the precise deterministic time, $T_2$, before which a
unique $L^2(\Omega)$-solution exits, but after which the series corresponding
to the $L^2(\Omega)$ moment of the solution blows up. By properly choosing the
parameters, results in this paper interpolate the known results for both
stochastic heat and wave equations.
