Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations
Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type,∂tβut(x)=−ν(−Δ)α/2ut(x)+I1−β[b(u)+σ(u)F⋅(t,x)] in (d + 1) dimensions, where ν > 0,β ∈ (0, 1), α ∈ (0, 2]. The operator ∂tβ is the Caputo fractional derivative while − (−Δ)α/2 is the generator of an isotropic α-stable Lévy process and I1−β is the Riesz fractional integral operator. The forcing noise denoted by 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.
Asogwa, S. A., Mijena, J. B., & Nane, E. (2020). Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations. Potential Analysis, 53(2), 357-386.