Title

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

Document Type

Article

Publication Date

2020

Publication Title

Potential Analysis

Abstract

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.

Department

Mathematics

Volume Number

53

Issue Number

2

First Page

357

Last Page

386

Comments

© 2019, This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply.

DOI

10.1007/s11118-019-09772-0

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