Boundedness, Ultracontractive Bounds and Optimal Evolution of the Support for Doubly Nonlinear Anisotropic Diffusion

Abstract

We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \begin{align*} \partial_t \big(|u|^{α-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 $t\mapsto |u{|}^{\alpha -1}u(\cdot ,t)\in L_{loc}^{1+1/\alpha }(\Omega )$ permitting us to use a suitable mollified weak formulation along with an appropriate test function.