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Nonlinear stochastic time-fractional diffusion equations on $ \mathbb{R}$: Moments, Hölder regularity and intermittency

Author: Le Chen
Journal: Trans. Amer. Math. Soc. 369 (2017), 8497-8535
MSC (2010): Primary 60H15; Secondary 60G60, 35R60
Published electronically: May 30, 2017
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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.

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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

Keywords: Nonlinear stochastic time-fractional diffusion equations, Anderson model, measure-valued initial data, H\"older continuity, intermittency, two-parameter Mainardi function
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).
Article copyright: © Copyright 2017 American Mathematical Society

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