On the initial boundary-value problem for \(u_ t-\text{div}(| \nabla u| ^{p-2}\nabla u)=0\)
type: Item
instance of: scholarly article
title: On the initial boundary-value problem for \(u_ t-\text{div}(| \nabla u| ^{p-2}\nabla u)=0\)
zbMATH Open document ID: 0831.35092
DOI: 10.1007/BF00375019
author: Xiangsheng Xu
published in: Archive for Rational Mechanics and Analysis
publication date: 1996-02-15T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^N\), \(N \geq 2\), be a bounded domain with Lipschitz boundary \(\partial \Omega\) and let \(T \in (0, \infty)\). The author considers the problem \( {\partial u / \partial t} - \text{div} (|\nabla u |^{p - 2} \nabla u) = 0\) in \(\Omega \times (0,T)\), \(u = 0\) on \(\partial \Omega \times (0,T)\), \(u(x,0) = u_0(x) \in L^1 (\Omega)\), \(1 < p < 2\). To define a suitable notion of a weak solution, he employs the idea of renormalization, first introduced by \textit{R. Di Perna} and \textit{P. L. Lions} [Commun. Pure Appl. Math. 42, No. 6, 729-757 (1989; Zbl 0698.35128)]. The author proves that there exists a unique renormalized solution.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 754019
zbMATH Keywords: renormalized solution
MaRDI profile type: Publication
Wikidata QID: Q114234137
cites work: Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when \(1<p<2\)
cites work: Global weak solutions of Vlasov‐Maxwell systems
cites work: Q3700884
description: scientific article; zbMATH DE number 754019