Moment bounds for a class of fractional stochastic heat equations

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}.

References in corpus (3)

• Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density
• On some properties of a class of fractional stochastic heat equations
• Non-linear noise excitation and intermittency under high disorder