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

Abstract

We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left({\partial }^{\alpha }-\text{B}\right)u(t,x)=u(t,x)\cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2,1)\cup (1,2);$$ $\text{B}$ 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.