Moment bounds for a class of fractional stochastic heat equations

Mohammud Foondun, Wei Liu, McSylvester Omaba

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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 languageEnglish
Pages (from-to)1-24
Number of pages24
JournalAnnals of Probability
Publication statusAccepted/In press - 19 Sep 2014

Keywords

  • intermittence
  • stochastic partial differential equations

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