  In the present paper, we study the existence and blow-up behavior to the
following stochastic non-local reaction-diffusion equation:
  \begin{equation*}
  \left\{
  \begin{aligned}
  du(t,x)&=\left[(\Delta+\gamma) u(t,x)+\int_{D}u^{q}(t,y)dy
-ku^{p}(t,x)+\delta u^{m}(t,x)\int_{D}u^{n}(t,y)dy \right]dt
  &\quad+\eta u(t,x)dB^{H}(t),
  u(t,x)&=0, \ \ t>0, \ \ x\in \partial D,
  u(0,x)&=f(x) \geq 0, \ \ x\in D,
  \end{aligned}
  \right.
  \end{equation*}
  where $D\subset \mathbb{R}^{d}\ (d \geq 1)$ is a bounded domain with smooth
boundary $\partial D$. Here, $k>0, \gamma, \delta, \eta \geq 0$ and $p,q,n>1,\
m\geq 0$ with $m+n \geq q\geq p$. The initial data $f $ is a non-negative
bounded measurable function in class $C^{2}$ which is not identically zero.
Here, $\left\{ B^{H}(t) \right\}_{t \geq 0} $ is a one-dimensional fractional
Brownian motion with Hurst parameter $\frac{1}{2} \leq H<1$ defined on a
filtered probability space $\left( \Omega, \mathcal{F}, (\mathcal{F}_{t})_{t
\geq 0}, \mathbb{P} \right)$. First, we estimate a lower bound for the
finite-time blow-up and by choosing a suitable initial data, we obtain the
upper bound for the finite-time blow-up of the above equation. Next, we provide
a sufficient condition for the global existence of a weak solution of the above
equation. Further, we obtain the bounds for the probability of blow-up
solution.
