  The $p$-Laplacian evolution equation in metric measure spaces has been
studied as the gradient flow in $L^2$ of the $p$-Cheeger energy (for $1 < p <
\infty$). In this paper, using the first-order differential structure on a
metric measure space introduced by Gigli, we characterize the subdifferential
in $L^2$ of the $p$-Cheeger energy. This gives rise to a new definition of the
$p$-Laplacian operator in metric measure spaces, which allows us to work with
this operator in more detail. In this way, we introduce a new notion of
solutions to the $p$-Laplacian evolution equation in metric measure spaces. For
$p = 1$, we obtain a Green-Gauss formula similar to the one by Anzellotti for
Euclidean spaces, and use it to characterise the $1$-Laplacian operator and
study the total variation flow. We also study the asymptotic behaviour of the
solutions of the $p$-Laplacian evolution equation, showing that for $1 \leq p <
2$ we have finite extinction time.
