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

TL;DR

This paper studies the regularity, boundedness, and support evolution of solutions to a class of doubly nonlinear anisotropic diffusion equations, introducing new methods linking energy class membership to solution regularity.

Contribution

It introduces a novel paradigm connecting energy class membership with regularity properties, including boundedness and support evolution, for doubly nonlinear anisotropic diffusion equations.

Findings

Established super and ultracontractive bounds for solutions.

Proved global boundedness and nonincreasing mass over time.

Showed compact support evolution for optimal exponents.

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|^{\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.