  We construct viscosity solutions to the nonlinear evolution equation
\eqref{p} below which generalizes the motion of level sets by mean curvature
(the latter corresponds to the case $p = 1$) using the regularization scheme as
in \cite{ES1} and \cite{SZ}. The pointwise properties of such solutions, namely
the comparison principles, convergence of solutions as $p\to 1$, large-time
behavior and unweighted energy monotonicity are studied. We also prove a
notable monotonicity formula for the weighted energy, thus generalizing
Struwe's famous monotonicity formula for the heat equation ($p =2$).
