Some non-existence results for a class of stochastic partial differential equations

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.