Hölder regularity of the nonlinear stochastic time-fractional slow and fast diffusion equations on ℝ^d

2021/05/03 by Chen, Le, Hu, Guannan
#35R60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 60H15. Secondary 60G60 #Probability (math.PR)

paper · doi:10.48550/arxiv.2105.00891

Abstract

In this paper, we use local fraction derivative to show the Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: (∂^β+\fracν2(-Δ)^α/2)u(t,x) = I_t^γ[σ(u(t,x))W(t,x)], tgt;0, x∈ℝ^d, where W is the space-time white noise, α∈(0,2], β∈(0,2), γ≥ 0 and ν>0, under the condition that 2(β+γ)-1-dβ/α>0. The case when β+γ≤ 1 has been obtained in \citeChHuNu19. In this paper, we have removed this extra condition, which in particular includes all cases for β∈(0,2).

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