Global existence and non-existence of weak solutions for non-local stochastic semilinear reaction–diffusion equations driven by a fractional noise

In this paper, we study the existence and blow-up behavior to the following stochastic non-local reaction–diffusion equation: [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary [Formula: see text]. Here, [Formula: see text] [Formula: see text] and [Formula: see text] with [Formula: see text]. The initial data [Formula: see text] is a non-negative bounded measurable function in class [Formula: see text] that is not identically zero. Here, [Formula: see text] is a one-dimensional fractional Brownian motion with Hurst parameter [Formula: see text] defined on a filtered probability space [Formula: see text] satisfying the usual conditions. First, we estimate a lower bound for the finite-time blow-up, and by choosing suitable initial data, we obtain an 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. Furthermore, we obtain the bounds for the probability of a blow-up solution.