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We prove the existence and uniqueness of mild solution for the stochasticpartial 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 byfractional, Riesz and Bessel kernel.

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