  Consider the following stochastic partial differential equation,
\begin{equation*} \partial_t u_t(x)=
  \mathcal{L}u_t(x)+ \sigma (u_t(x))\dot F(t,x)\quad{t>0}\quad\text{and}\quad
x\in R^d. \end{equation*} The operator $\mathcal{L}$ is the generator of a
strictly stable process and $\dot F$ is the random forcing term which is
assumed to be Gaussian. Under some additional conditions, most notably on
$\sigma$ and the initial condition, we show non-existence of global random
field solutions. Our results are new and complement earlier works.
