Generalized Hardy-Sobolev inequalities and exponential decay of the entropy of \(g(x)\dot u=\Delta u\)
type: Item
instance of: scholarly article
title: Generalized Hardy-Sobolev inequalities and exponential decay of the entropy of \(g(x)\dot u=\Delta u\)
zbMATH Open document ID: 1058.35131
author: Mythily Ramaswamy
author: A. Unterreiter
published in: Monatshefte für Mathematik
publication date: 2004-11-05T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^d \) be a smooth bounded domain and \(g \in L^1_{\text{loc}}(\Omega)\) be a non-negative function which satisfies the following generalized Hardy-Sobolev inequality: \[ \int_{\Omega} v^2 g \,dx \leq C \int_{\Omega} | \nabla v| ^2 \,dx,\quad \forall v \in H^1_o(\Omega), tag 1 \] where \(C \in \mathbb{R}^+\) is a constant which only depends on \(\Omega.\) The authors consider the (possibly degenerated) parabolic equation \(g \frac{\partial u}{\partial t} = \Delta u \) in \((0,t) \times \Omega,\) associated with the homogeneous Dirichlet boundary conditions on the boundary of \(\Omega,\) and the initial condition \(u| _{t=0} =u_0 \in L^2 (g \,dx)\). They prove the existence and the uniqueness of a weak solution of this problem and the validity of the maximum-minimum principle. They study the behavior of the associated entropy \(E(t),\) as \(t \to \infty,\) and show the exponential decay law \(E(t)\leq E(0) e^{-\frac{2t}{C}},\) where \(C\) is the constant in (1). An analogous exponential decay of the square gradient norm is also obtained. The main tools in the proofs, are the properties of the Lebesgue-measure, of the Lebesgue-Sobolev spaces and of the L. Schwartz 's theory of distributions
reviewed by: Denise Huet
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2113920
zbMATH Keywords: maximum principle
zbMATH Keywords: minimum principle
zbMATH Keywords: weak solution
MaRDI profile type: Publication
full work available at URL: s00605-003-0041-6
OpenAlex ID: W2058321891
DOI: 10.1007/S00605-003-0041-6
description: scientific article; zbMATH DE number 2113920