  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.
