  We consider a class of porous medium type of equations with Caputo time
derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a
bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet
boundary conditions. The operator $\A$ 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 $m>1$. This equation is the
most general form of a variety of models used to describe anomalous diffusion
processes with memory effects, and finds application in various fields,
including visco-elastic materials, signal processing, biological systems and
geophysical science.
  We show existence of unique solution and new $L^p-L^\infty$ smoothing
effects. 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. Consequently, we establish that the regularizing effects
from the diffusion are stronger than the memory effects introduced by the
fractional time derivative. Moreover, the solution attains the boundary
conditions pointwise.
  Finally, we prove that the 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 memory effects weaken the spatial diffusion and mitigate
the difference between slow and fast diffusion.
