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

Abstract

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$ \partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)] $$ in (d + 1) dimensions, where ν > 0,β ∈ (0, 1), α ∈ (0, 2]. The operator $\partial ^{\beta }_{t}$ is the Caputo fractional derivative while − (−Δ)α/2 is the generator of an isotropic α-stable Levy process and I1−β is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel {\cdot }{F}(t,x)$ 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, σ 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.