Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Consider the stochastic partial differential equation ∂ ∂ t u t ( x ) = − ( − Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where − ( − Δ ) α 2 -(-\Delta )^{\frac {\alpha }{2}} denotes the fractional Laplacian with power α 2 ∈ ( 1 2 , <