On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem
type: Item
instance of: scholarly article
title: On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem
zbMATH Open document ID: 1050.35154
author: J. L. Da Silva
author: José Francisco Rodrigues
published in: Communications on Pure and Applied Analysis
publication date: 2004-05-27T00:00:00Z
review text: The authors start by considering the reaction-diffusion system \[ \partial_t u-\nabla \cdot (a(\theta)\nabla u)=f\quad\text{ in }\{u > 0\} \] \[ \tau \partial_t \theta - \sigma \Delta \theta +\theta =\chi_{\{u>0\}}\quad\text{ in }Q=\Omega \times (0,T) \] with \(\Omega\) a bounded domain in \(\mathbb R^n\), where, to fix ideas, \(u\) is a concentration and \(\theta\) is temperature. The diffusivity \(a(\theta)\) is a strictly positive continuous function. The unknown \(u(t)\) is sought in the convex set \(K=\{v\in H^1_0(\Omega): v>0\) in \(\Omega\}\) and \(\theta\) satisfies a homogeneous Neumann condition. The initial data \(u_0,\theta_0\) are such that \(u_0\in K\), \(\theta_0\in W^{2,\infty}(\Omega)\) and \(\partial_{\nu}\theta_0=0\quad\text{ on } \partial \Omega\). Under the assumptions \(\partial \Omega \in C^2\), \(f \in L^2(Q), a(\theta)\) Lipschitz, \(f\neq 0\) a.e. in \(Q\), existence is proved for any pair \((\tau, \sigma)\) of positive constants, with the property \(u_{\tau \sigma}\in W^{2,1}_2 (Q)\), \(\theta_{\tau \sigma}\in W^{2,1}_q(Q)\), \(\nabla q<\infty\). The problem is formulated in a variational form and a fixed point argument is used. Next the authors investigate the limit \(\sigma \rightarrow +\infty\) (Problem \(P_{\tau}\)) and the further limit \(\tau \rightarrow 0\). In \(P_{\tau}\) the equation for the temperature is replaced by \[ \tau \dfrac{d\xi}{d\tau}+\xi =\langle u^+ \rangle, \] \(\langle u^+\rangle\) representing the volume fraction of the set \(\{u>0\}\). A solution \(u_\tau \in W^{2,1}_2 (Q)\), \(\xi_\tau \in W^{1,\infty}(0,T)\) is obtained as an appropriate limit of \(u_{\tau \sigma}\theta _{\tau \sigma}\). A discussion is performed about the assumption \(f\neq 0\) in \(Q\) and on how to reformulate the problem in its absence, still preserving existence. Finally, letting \(\tau \rightarrow 0\), basically \(\xi\) is replaced by \(\langle u^+\rangle\) and Problem P consists in just one variational inequality, which again is shown to…
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35K85
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 2066963
zbMATH Keywords: variational inequalities
zbMATH Keywords: reaction-diffusion system
zbMATH Keywords: bounded domain
zbMATH Keywords: existence
zbMATH Keywords: variational form
zbMATH Keywords: fixed point argument
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: cpaa.2004.3.85
OpenAlex ID: W2073572953
DOI: 10.3934/CPAA.2004.3.85
description: scientific article; zbMATH DE number 2066963