  We consider fractional stochastic heat equations of the form $\frac{\partial
u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot
F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions,
we show that the second moment of the solution grows exponentially with time.
In particular, this answers an open problem in \cite{CoKh}. Along the way, we
prove a number of other interesting properties which extend and complement
results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and
\cite{Khoshnevisan:2013ab}.
