Nonlinear stochastic time-fractional diffusion equations on $\mathbb {R}$: Moments, Hölder regularity and intermittency
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Abstract:
We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\mathbb {R}$, driven by multiplicative space-time white noise. The fractional index $\beta$ varies continuously from $0$ to $2$. The case $\beta =1$ (resp. $\beta =2$) corresponds to the stochastic heat (resp. wave) equation. The cases $\beta \in \:]0,1[\:$ and $\beta \in \:]1,2[\:$ are called slow diffusion equations and fast diffusion equations, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all $p$-th moments $(p\ge 2)$ are obtained, which are expressed using a kernel function $\mathcal {K}(t,x)$. The second moment is sharp. We obtain the Hölder continuity of the solution for the slow diffusion equations when the initial data is a bounded function. We prove the weak intermittency for both slow and fast diffusion equations. In this study, we introduce a special function, the two-parameter Mainardi functions, which are generalizations of the one-parameter Mainardi functions.References
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Additional Information
- Le Chen
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Address at time of publication: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045
- MR Author ID: 1076493
- ORCID: 0000-0001-8010-136X
- Email: chenle02@gmail.com, chenle@ku.edu
- Received by editor(s): October 8, 2014
- Received by editor(s) in revised form: January 23, 2016
- Published electronically: May 30, 2017
- Additional Notes: This research was supported both by the University of Utah and by a fellowship from the Swiss National Foundation for Scientific Research (P2ELP2_151796).
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8497-8535
- MSC (2010): Primary 60H15; Secondary 60G60, 35R60
- DOI: https://doi.org/10.1090/tran/6951
- MathSciNet review: 3710633