Harnack's inequality of weak type for the parabolic \(p (x)\)-Laplacian
type: Item
instance of: scholarly article
title: Harnack's inequality of weak type for the parabolic \(p (x)\)-Laplacian
zbMATH Open document ID: 1486.35009
DOI: 10.1134/S0001434622010199
author: M. D. Surnachev
published in: Mathematical Notes
publication date: 2022-03-14T00:00:00Z
review text: In this short but very interesting paper, the author establishes Harnack's inequality of weak type for nonnegative bounded supersolutions of the parabolic \(p(x)\)-Laplacian \[ u_t-\operatorname{div} A(x, t, \nabla u)=0\text{ in }Q := D \times [a, b], \] where \(D\) is a bounded Lipschitz domain in \(\mathbb{R}^n\), \(n\geq2\), \(a<b\). The flow \(A\) is a Caratheodory function and satisfies the coercivity and growth conditions \[ A(x, t, \xi)\cdot\xi\geq C_0|\xi|^{p(x)} \] \[ |A(x, t,\xi)|\leq C_1|\xi|^{p(x)-1} \] for all \(\xi\in R^n\) and almost all $(x, t)\in Q$, where \(C_0\) and \(C_1\) are positive constants. The variable exponent \(p\in L^\infty(D)\) is bounded away from 1 and $\infty$ and \(1 < p_1\leq p(x) \leq p_2 <\infty\) for almost all \(x\in D\). The author supposes that the exponent $p(\cdot)$ satisfies the Zhikov's logarithmic condition \[ |p(x)- p(y)|\leq L\frac{1}{ \ln(e + r-1)}\,. \] Under such assumptions, the author proves Harnack's inequality of weak type for nonnegative bounded supersolutions.
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35A23
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K92
zbMATH DE Number: 7488491
zbMATH Keywords: Harnack inequality
zbMATH Keywords: \((p (x))\)-Laplacian
zbMATH Keywords: nonnegative supersolutions
MaRDI profile type: Publication
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full work available at URL: s0001434622010199
OpenAlex ID: W4213424211
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description: scientific article; zbMATH DE number 7488491