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

2023/06/29 by Ciani, Simone, Vespri, Vincenzo, Vestberg, Matias · 2 citations
#35B45 #35B65 #35D30 #35K10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.17152

Abstract

We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is ∂_t (|u|^α-1u ) - ∑^N_i=1 ∂_i ( |∂_i u|^{p_i - 2} ∂_i u ) = 0, where α∈ (0,1) 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-1u(⋅,t) ∈ L^1+1/α_loc(Ω) permitting us to use a suitable mollified weak formulation along with an appropriate test function.

Cited by

• Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations
• Existence, comparison principle and uniqueness for fully nonlinear anisotropic evolution equations

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