We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \(\begin{aligned} \partial _t \big (|u|^{\alpha -1}u \big ) - \sum ^N_{i=1} \partial _i \big ( |\partial _i u|^{p_i - 2} \partial _i u \big ) = 0, \end{aligned}\) where \(\alpha > 0\) 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 \(t \mapsto |u|^{\alpha -1}u(\cdot ,t) \in L^{1+1/\alpha }_{\text {loc}}(\Omega )\) permitting us to use a suitable mollified weak formulation along with an appropriate test function.
