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 (Formula presented.) where α>0 and pi∈(1,∞). We obtain super and ultracontractive bounds, and global boundedness in space for solutions to the Cauchy problem with initial data in Lα+1(RN), 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-1u(·,t)∈Lloc1+1/α(Ω) permitting us to use a suitable mollified weak formulation along with an appropriate test function.