Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

Le Chen, Yaozhong Hu and David Nualart

Stochastic Processes and their Applications, 2019, vol. 129, issue 12, 5073-5112

Abstract: This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: ∂β+ν2(−Δ)α∕2u(t,x)=Itγρ(u(t,x))Ẇ(t,x),t>0,x∈Rd,where Ẇ is the space–time white noise, α∈(0,2], β∈(0,2), γ≥0 and ν>0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: d<2α+αβmin(2γ−1,0). In some cases, the initial data can be measures. When β∈(0,1], we prove the sample path regularity of the solution.

Keywords: Nonlinear stochastic fractional diffusion equations; Measure-valued initial data; Hölder continuity; Intermittency; The Fox H-function (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2019.01.003

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