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We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account "memory''. The precise model is   \[   D_t^{\alpha} u - \text{div}(u(-\Delta)^{-\sigma} u) = f, \quad 0<\sigma <1/2.   \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0 $ as well as right hand side $f\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is H\"older continuity for such weak solutions.

Porous medium flow with both a fractional potential pressure and fractional time derivative