25. chen.hu:22:holder

Hölder regularity for the nonlinear stochastic time-fractional slow & fast diffusion equations on \(\mathbb{R}^d\)

Le Chen and Guannan Hu

Abstract In this paper, we study the space-time Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation:

\[\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) = I_t^\gamma\left[\sigma\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\ x\in\mathbb{R}^d,\]

where \(\dot{W}\) is the space-time white noise, \(\alpha\in(0,2]\), \(\beta\in(0,2)\), \(\gamma\ge 0\) and \(\nu>0\). The existence/uniqueness of a random field solution has been obtained in [CHN19] under the condition that \(2(\beta+\gamma)-1-d\beta/\alpha>0\). The Hölder regularity of the solution has been obtained in the same reference, but only for the case \(\beta+\gamma\le 1\). In this paper, we use the idea from the local fractional derivative to establish the Hölder regularity of the solution for all possible cases – \(\beta\in(0,2)\), which in particular recovers the special case in [CHN19] when \(\beta\in (0,1-\gamma]\). As a rather surprising consequence, when \(\gamma=0\), \(\alpha=2\) and \(\beta\) is close to \(2\), the space and time Hölder exponents are both to \(1-\), which is different from the known Hölder exponents for the stochastic wave equation which are \((1/2)-\).

MSC 2010 subject classifications: Primary. 60H15; Secondary. 60G60, 26A33, 60F05.

Keywords: nonlinear stochastic time-fractional slow and fast diffusion equations, local fractional derivative, fractional Taylor theorem, Hölder continuity, Mittag-Leffler function, Fox H-function.

Downlaod

[CH22] Chen, Le & Hu, Guannan (2022) ‘H”older regularity for the nonlinear stochastic time-fractional slow & fast diffusion equations on \(\mathbb{R}^d\)’, Fract. Calc. Appl. Anal. 25, 608–629. <https://doi.org/10.1007/s13540-022-00033-3>

@article{chen.hu:22:holder,
  author        = {Chen, Le and Hu, Guannan},
  title         = {H\"{o}lder regularity for the nonlinear stochastic time-fractional slow \& fast diffusion equations on {$\Bbb R^d$}},
  journal       = {Fract. Calc. Appl. Anal.},
  fjournal      = {Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications},
  volume        = {25},
  year          = {2022},
  number        = {2},
  pages         = {608--629},
  issn          = {1311-0454},
  mrclass       = {60H15 (26A33 35R11)},
  mrnumber      = {4437294},
  doi           = {10.1007/s13540-022-00033-3},
  url           = {https://doi.org/10.1007/s13540-022-00033-3}
}

References: Ben Adda and Cresson [BAC13]; Chen [Che13]; Chen [Che17a]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen et al. [CHN19]; Chen et al. [CHHH17]; Chen et al. [CYZ10]; Chen et al. [CKK15]; Dalang [Dal99]; Erdélyi et al. [EMOT81b]; Foondun and Khoshnevisan [FK09]; Hu and Hu [HH15]; Kilbas and Saigo [KS04]; Kilbas et al. [KST06]; Kolwankar and Gangal [KG96]; Kolwankar and Gangal [KG98]; Liu et al. [LRdS18]; Mijena and Nane [MN15]; Podlubny [Pod99]; Samko et al. [SKM93]; Schneider [Sch96]; Talvila [Tal01]; Walsh [Wal86];

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