  Consider non-linear time-fractional stochastic reaction-diffusion equations
of the following type, $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2}
u_t(x)+I^{1-\beta}_t[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$
dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator
$\partial^\beta_t$ is the Caputo fractional derivative while
$-(-\Delta)^{\alpha/2} $ is the generator of an isotropic $\alpha$-stable
L\'evy process and $I^{1-\beta}_t$ is the Riesz fractional integral operator.
The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise.
These equations might be used as a model for materials with random thermal
memory. We derive non-existence (blow-up) of global random field solutions
under some additional conditions, most notably on $b$, $\sigma$ and the initial
condition. Our results complement those of P. Chow in \cite{chow2},
\cite{chow1}, and Foondun et al. in \cite{Foondun-liu-nane},
\cite{foondun-parshad} among others.
