We consider a class of porous medium type equations with Caputo time derivative. The prototype problem reads as \(D^\alpha _t u=-\mathcal {L}u^m\) and is posed on a bounded Euclidean domain \(\Omega \subset \mathbb {R}^N\) with smooth boundary and zero Dirichlet boundary conditions. The linear operator \(\mathcal {L}\) falls within a wide class of either local or nonlocal operators, and the nonlinearity is allowed to be of degenerate or singular type, namely, \(0<m<1\) and \(m>1\) . Using the fractional gradient flow theory, we show existence of unique solution and prove new \(L^p-L^\infty \) smoothing effects by employing the Green function method. The comparison principle, which we provide in the most general setting, serves as a crucial tool in the proof and provides a novel monotonicity formula. To the best of our knowledge, this type of smoothing effects are novel in nonlinear PDEs involving Caputo derivative. Consequently, we establish that the regularizing effect due to the diffusion is stronger than the memory effect introduced by the fractional time derivative. We also prove that the solution satisfies the zero Dirichlet condition pointwise by means of sharp boundary estimates. Finally, we prove that our solution does not vanish in finite time if \(0<m<1\) , unlike the case with the classical time derivative. Indeed, we provide a sharp rate of decay for any \(L^p\) -norm of the solution for any \(m>0\) . Our findings indicate that the difference between slow and fast diffusion is mitigated by the memory effect, which dramatically slows the spatial diffusion.
