Boundedness, ultracontractive bounds and optimal evolution of the support for doubly nonlinear anisotropic diffusion

We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is $$∂_t(|u|^{α−1}u−\Sum^N_{i=1)∂_i(|∂_i u|^{p_i−2)∂_i u = 0$$, where α > 0 and p_i ∈ (1,∞). We obtain super and ultracontractive bounds, and global boundedness in space for solutions to the Cauchy problem with initial data in $L^{α+1}(R^N )$, 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 $t → |u|^{α−1}u(·, t) ∈ L^{1+1/α}_{loc} (\Omega)$ permitting us to use a suitable mollified weak formulation along with an appropriate test function.