  In this paper, we investigate the existence and finite-time blow-up for the
solution of a reaction-diffusion system of semilinear stochastic partial
differential equations (SPDEs) subjected to a two-dimensional fractional
Brownian motion given by
  \begin{eqnarray*}
  du_{1}(t,x)&=&\left[
\Delta_{\alpha}u_{1}(t,x)+\gamma_{1}u_{1}(t,x)+u^{1+\beta_{1}}_{2}(t,x)
\right]dt
  &\qquad \ \ +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t),
  du_{2}(t,x)&=&\left[
\Delta_{\alpha}u_{2}(t,x)+\gamma_{2}u_{2}(t,x)+u^{1+\beta_{2}}_{1}(t,x)
\right]dt
  &\qquad \ \ +k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t),
  \end{eqnarray*}
  for $x \in \mathbb{R}^{d},\ t \geq 0$, along with \begin{equation*}
  \begin{array}{ll}
  u_{i}(0,x)=f_{i}(x), &x \in \mathbb{R}^{d}, \nonumber
  \end{array}
  \end{equation*}
  where $\Delta_{\alpha}$ is the fractional power
$-(-\Delta)^{\frac{\alpha}{2}}$ of the Laplacian, $0<\alpha \leq 2$ and
$\beta_{i}>0,\ \gamma_{i}>0$ and $k_{ij}\geq 0, i,j=1,2$ are constants. We
provide sufficient conditions for the existence of a global weak solution.
Under the assumption that $\beta_{1}\geq \beta_{2}>0$ with Hurst index $ 1/2
\leq H < 1,$ we obtain the blow-up times for an associated system of random
partial differential equations in terms of an integral representation of
exponential functions of Brownian motions. Moreover, we provide lower and upper
bounds for the finite-time blow-up of the above system of SPDEs and obtain the
upper bounds for the probability of non-explosive solutions to our considered
system.
