Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, partial derivative(beta)(t)u(t)(x)=-nu(-Delta)(alpha/2)u(t)(x)+I1-beta[b(u)+sigma(u)(F) over dot (t,x)] in (d + 1) dimensions, where nu > 0, beta is an element of (0, 1), alpha is an element of (0, 2]. The operator partial derivative(beta)(t) is the Caputo fractional derivative while - (-Delta)(alpha/2) is the generator of an isotropic alpha-stable Levy process and I1-beta is the Riesz fractional integral operator. The forcing noise denoted by (F) over dot (t,x) is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on b, sigma and the initial condition. Our results complement those of P. Chow in (Commun. Stoch. Anal. 3(2):211-222,2009), Chow (J. Differential Equations 250(5):2567-2580, 2011), and Foondun et al. in (2016), Foondun and Parshad (Proc. Amer. Math. Soc.143(9):4085-4094, 2015) among others.