Work

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

Year: 2025

Type: article

Abstract: We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is $$\begin{aligned} \partial _t \big (|u|^{\alpha -1}u \big ) - \sum ^N_{i=1} \partial _i \big ( |\partial _i u|^{p_i - 2} \partial _i u \big ) = 0, \end{aligned}$$ ∂ t ( | u | α - 1 u ) - ∑ i = 1 N ∂ i ( | ∂ i u | p i - 2 ∂ i u ) = 0 , where $$\alpha > 0$$ α > 0 and $$p_i \in (1, \infty )$$ p i ∈ ( 1 , ∞ ) . We obtain super and ultracontractive bounds, and global boundedness in space for solutions to the Cauchy problem with initial data in $$L^{\alpha +1}(\mathbb {R}^N)$$ 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 $$t \mapsto |u|^{\alpha -1}u(\cdot ,t) \in L^{1+1/\alpha }_{\text {loc}}(\Omega )$$ t ↦ | u | α - 1 u ( · , t ) ∈ L loc 1

Source: Annali di Matematica Pura ed Applicata (1923 -)

DOI: 10.1007/s10231-025-01600-0

Authors: Simone Ciani, Vincenzo Vespri, Matias Vestberg

Institutions: University of Bologna, University of Florence, Uppsala University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190