Global and non-global solutions of a fractional reaction-diffusion equation perturbed by a fractional noise

We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation where and are constants, is the fractional power of the Laplacian, is a fractional Brownian motion with Hurst parameter H, and is a bounded measurable function. To achieve this we investigate the growth of integrals of the form as Moreover, we provide sufficient conditions for the existence of a global weak solution of the above equation, as well as upper and lower bounds for the probability that the solution does not blow up in finite time.