Some stochastic time-fractional diffusion equations with variable coefficients and time dependent noise

We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^\alpha - \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$ is an uniform elliptic operator with variable coefficients and $\dot W$ is a Gaussian noise general in time with space covariance given by fractional, Riesz and Bessel kernel.