On some contractive properties for the heat equations
type: Item
instance of: scholarly article
title: On some contractive properties for the heat equations
zbMATH Open document ID: 0722.35041
author: Tetsuya Koyama
published in: Hiroshima Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: For a solution u(x,t) of the initial-boundary value problem for the heat equation \[ (1)\quad \partial u/\partial t=\Delta u,\quad x\in \Omega,\quad t>0; \] \[ u(x,0)=u_ 0,\quad x\in \Omega;\quad u(x,t)=0,\quad x\in \partial \Omega,\quad t>0 \] where \(\Omega\) is a bounded domain in \(R^ N\), the following contraction property of the corresponding semigroup is considered: \[ (2)\quad \| \Delta u(\cdot,t)\|_{L^ p(\Omega)}\leq \| \nabla u(\cdot,s)\|_{L^ p(\Omega)},\quad 0<s\leq t, \] where \(1\leq p<\infty\), and \(\nabla\) is the gradient of u. Three functionals which are equivalent to the functional \(u\mapsto \max \{| u(x)-u(y)|\), (x,y)\(\in \Omega,| x-y| \leq \epsilon \}\), the usual Hölder norm and Lipschitz norm, respectively, and which represent the regularity of u, are introduced. It is shown that the above three functionals have the same type of contractive properties as in (2) under the assumption that \(\Omega\) is convex.
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 4190264
zbMATH Keywords: contraction property
zbMATH Keywords: Lipschitz norm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206129179
OpenAlex ID: W1556563057
Wikidata QID: Q128871898
DOI: 10.32917/HMJ/1206129179
description: scientific article; zbMATH DE number 4190264