We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).
- fractional laplacian
- heat kernel
- stochastic heat equation
- stochastic partial differential equations