7. chen:17:nonlinear

Nonlinear stochastic time-fractional diffusion equations on** \(\mathbb{R}\): moments, Hölder regularity and intermittency

Le Chen

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.

MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.

Keywords: nonlinear stochastic time-fractional diffusion equations, Anderson model, measure-valued initial data, Hölder continuity, intermittency, two-parameter Mainardi function.

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[Che17a] Chen, Le (2017) ‘Nonlinear stochastic time-fractional diffusion equations on \(\mathbb{R}^d\): moments, H”older regularity and intermittency’, Trans. Amer. Math. Soc. 369, 8497–8535. <https://doi.org/10.1090/tran/6951>

@article{chen:17:nonlinear,
  author        = {Chen, Le},
  title         = {Nonlinear stochastic time-fractional diffusion equations on {$\Bbb{R}$}: moments, {H}\"{o}lder regularity and intermittency},
  journal       = {Trans. Amer. Math. Soc.},
  fjournal      = {Transactions of the American Mathematical Society},
  volume        = {369},
  year          = {2017},
  number        = {12},
  pages         = {8497--8535},
  issn          = {0002-9947},
  mrclass       = {60H15 (35R11 35R60 60G60)},
  mrnumber      = {3710633},
  mrreviewer    = {Feng-Yu Wang},
  doi           = {10.1090/tran/6951},
  url           = {https://doi.org/10.1090/tran/6951}
}

References: Aki and Richards [AR09]; Bertini and Cancrini [BC95a]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen and Kim [CK17]; Conus et al. [CJKS13a]; Conus et al. [CJKS14]; Craiem et al. [CRA+08]; Debbi and Dozzi [DD05]; Diethelm [Die10]; Dimitrienko [Dim11]; Doi and Edwards [DE86]; Ferry [Fer80]; Foondun and Khoshnevisan [FK09]; Kardar et al. [KPZ86]; Klausner [Kla12]; Lemaitre and Jean-Louis [LJL90]; Magin [Mag10]; Mainardi [Mai10]; Mainardi et al. [MLP01]; Mijena and Nane [MN15]; Mijena and Nane [MN16]; Oldham et al. [OMS09]; Olver et al. [OLBC10]; Perkins and Lach [PL11]; Podlubny [Pod99]; Polyanin [Pol02]; Schneider [Sch96]; Umarov and Saydamatov [US06]; Walsh [Wal86]; Widder [Wid41]; Wright [Wri40]; Wright [Wri35];

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