On some properties of a class of fractional stochastic heat equations

We consider nonlinear parabolic stochastic equations of the form $\partial_t u=\sL u + \lambda \sigma(u)\dot \xi$ontheball$B(0,,R)$,where$\dot \xi$denotessomeGaussiannoiseand$\sigma$isLipschitzcontinuous\mathrm{.}Here$\sL$correspondstoan$\alpha$-stableprocesskilleduponexiting$B(0, R)$. We will consider two types of noise; space-time white noise and spatially correlated noise. Under a linear growth condition on $\sigma$, we study growth properties of the second moment of the solutions.