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In this paper, we investigate the existence and finite-time blow-up for thesolution of a reaction-diffusion system of semilinear stochastic partialdifferential equations (SPDEs) subjected to a two-dimensional fractionalBrownian motion given by $$\left\{\begin{aligned} du_{1}(t,x)&=\left[\Delta_{\alpha}u_{1}(t,x)+\gamma_{1}u_{1}(t,x)+u^{1+\beta_{1}}_{2}(t,x)\right]dt +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t), \\du_{2}(t,x)&=\left[\Delta_{\alpha}u_{2}(t,x)+\gamma_{2}u_{2}(t,x)+u^{1+\beta_{2}}_{1}(t,x)\right]dt+k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t), \\\end{aligned}\right.$$ for $x \in \mathbb{R}^{d},\ t \geq 0$, along with$$\begin{array}{ll} u_{i}(0,x)=f_{i}(x), &x \in \mathbb{R}^{d}, \nonumber\\\end{array}$$ 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 $k_{ij}\neq 0, i,j=1,2$ are constants. Weprovide 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 randompartial differential equations in terms of an integral representation ofexponential functions of Brownian motions. Moreover, we provide lower and upperbounds for the finite-time blow-up of the above system of SPDEs and obtain thelower and upper bounds for the probability of non-explosive solutions to ourconsidered system.

Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise