Summary: The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional and is of Caputo-type, which takes into account “memory”. The precise model is \[ D_t^\alpha u - \operatorname{div}(u(- \Delta)^{- \sigma}u) = f,\quad 0 < \sigma < \frac{1}{2}. \] This paper poses the problem over \(\{t \in \mathbb R^{+}, x \in \mathbb R^{n}\}\) with nonnegative initial data \(u(0,x) \geq 0\) as well as the right-hand side \(f \geq 0\). The existence for weak solutions when \(f,u(0,x)\) have exponential decay at infinity is proved. The main result is Hölder continuity for such weak solutions.