### Abstract

We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Annals of Probability |

Publication status | Accepted/In press - 19 Sep 2014 |

### Keywords

- intermittence
- stochastic partial differential equations

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## Cite this

Foondun, M., Liu, W., & Omaba, M. (Accepted/In press). Moment bounds for a class of fractional stochastic heat equations.

*Annals of Probability*, 1-24.