Blow-up results for space-time fractional stochastic partial differential equations

2018/03/15 by Asogwa, Sunday, Mijena, Jebessa B., Nane, Erkan · 1 citation
#35B44 #35R60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1803.05890

Abstract

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, ∂^β_tu_t(x)=-ν(-Δ)^α/2 u_t(x)+I^1-β_t[b(u)+ σ(u)\stackrel⋅F(t,x)] in (d+1) dimensions, where ν>0, β∈ (0,1), α∈ (0,2]. The operator ∂^β_t is the Caputo fractional derivative while -(-Δ)^α/2 is the generator of an isotropic α-stable Lévy process and I^1-β_t is the Riesz fractional integral operator. The forcing noise denoted by \stackrel⋅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 \citechow2, \citechow1, and Foondun et al. in \citeFoondun-liu-nane, \citefoondun-parshad among others.

Cited by

• Fractional Stochastic Navier-Stokes Equations: Local Well-Posedness and Enstrophy Balance

Related

• Asymptotic properties of some space-time fractional stochastic equations
• Positive effects of multiplicative noise on the explosion of nonlinear fractional stochastic differential equations
• Space-time fractional stochastic partial differential equations driven by Lévy white noise
• Global existence and non-existence of weak solutions for non-local stochastic semilinear reaction-diffusion equations driven by a fractional noise
• Pathwise Blowup of space-time fractional SPDEs