  We investigate some regularity properties of a class of doubly nonlinear
anisotropic evolution equations whose model case is \begin{align*}
  \partial_t \big(|u|^{\alpha -1}u \big) - \sum^N_{i=1} \partial_i \big(
|\partial_i u|^{p_i - 2} \partial_i u \big) = 0, \end{align*} where $\alpha \in
(0,1)$ and $p_i \in (1, \infty)$. We obtain super and ultracontractive bounds,
and global boundedness in space for solutions to the Cauchy problem with
initial data in $L^{\alpha+1}(\mathbb{R}^N)$, and show that the mass is
nonincreasing over time. As a consequence, compactly supported evolution is
shown for optimal exponents. We introduce a seemingly new paradigm, by showing
that Caccioppoli estimates, local boundedness and semicontinuity are
consequences of the membership to a suitable energy class. This membership is
proved by first establishing the continuity of the map $t \mapsto
|u|^{\alpha-1}u(\cdot,t) \in L^{1+1/\alpha}_{loc}(\Omega)$ permitting us to use
a suitable mollified weak formulation along with an appropriate test function.
