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

Abstract

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.