Weak Harnack inequality for doubly non-linear equations of slow diffusion type
type: Item
instance of: scholarly article
title: Weak Harnack inequality for doubly non-linear equations of slow diffusion type
zbMATH Open document ID: 1547.35124
author: Fabian Bäuerlein
published in: Journal of Mathematical Analysis and Applications
publication date: 2024-09-03T00:00:00Z
review text: In this paper, the author considers non-negative weak super-solutions \(u : \Omega_T \rightarrow R \geq 0\) to the doubly nonlinear equation \N\[\N\partial_t|u|^{q-1}u-\operatorname{div}A(x, t, u,Du) = 0\N\]\Nin \(\Omega_T = \Omega \times (0, T]\), where \(\Omega\) is an bounded open set in \(\mathbb{R}^N\) for \(N\geq 2\), \(T > 0\) and \(q\) is a non-negative parameter, and the vector field \(A\) satisfies standard \(p\)-growth assumptions for some \(p > 1\). The weak Harnack inequality is established in the entire slow diffusion regime \(p-q-1 > 0\) and this represents the real novelty of the paper. Lastly, it is worth to be noticed that the weak super-solution \(u\) is assumed only to belong to the function space\N\[\NC^0_{\mathrm{loc}}([0, T];L^{q+1}_{\mathrm{loc}}(\Omega))\cap L^p_{\mathrm{loc}}([0, T];W^{1,p}_{\mathrm{loc}}(\Omega)).\N\]
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 7905806
zbMATH Keywords: doubly nonlinear equations
zbMATH Keywords: super-solutions
zbMATH Keywords: weak Harnack inequality
zbMATH Keywords: expansion of positivity
MaRDI profile type: Publication
cites work: Regularity for minimizers of non-quadratic functionals: The case \(1<p<2\)
cites work: On the diffusive wave approximation of the shallow water equations
cites work: On the Hölder regularity of signed solutions to a doubly nonlinear equation
cites work: Harnack's inequality for doubly nonlinear equations of slow diffusion type
cites work: Local smoothing effects, positivity, and Harnack inequalities for the fast \(p\)-Laplacian equation
cites work: Local clustering of the non-zero set of functions in \(W^{1,1} (E)\)
cites work: Harnack estimates for quasi-linear degenerate parabolic differential equations
cites work: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
cites work: Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations
cites work: A boundary estimate for singular parabolic diffusion equations
cites work: Q5486108
cites work: Remarks on the regularity of the minimizers of certain degenerate functionals
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Q3151387
cites work: Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation
cites work: Local behaviour of solutions to doubly nonlinear parabolic equations
cites work: Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations
cites work: A weak Harnack estimate for supersolutions to the porous medium equation.
cites work: Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations
cites work: Instantaneous optimal control of friction dominated flow in a gas-network
cites work: Local boundedness of weak solutions to the diffusive wave approximation of the shallow water equations
cites work: On harnack type inequalities and their application to quasilinear elliptic equations
cites work: Pointwise estimates and quasilinear parabolic equations
cites work: Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations
cites work: Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations
cites work: An extensive study of the regularity of solutions to doubly singular equations
DOI: 10.1016/J.JMAA.2024.128541
description: scientific article; zbMATH DE number 7905806