Monotonicity formula for a problem with hysteresis
type: Item
instance of: scholarly article
title: Monotonicity formula for a problem with hysteresis
zbMATH Open document ID: 1401.35302
DOI: 10.1134/S1064562418010167
published in: Doklady Mathematics
publication date: 2018-06-20T00:00:00Z
review text: This short note states without proof a local monotonicity formula for the initial boundary value problem \[ \begin{aligned} \partial_t u - \Delta u + h[u] & = 0 \;\text{ in }\; (0,T)\times \mathcal{U}\;, \\ u(0) & = \varphi \;\text{ in }\; \mathcal{U}\;,\end{aligned} \] supplemented with either Dirichlet or Neumann boundary conditions. Here, \(h[u]\) is a discontinuous nonlinearity built upon a function \(f:\mathbb{R}\to \{-1,1\}\) and is defined as follows: \(h[u](t,x)=f(u(t,x))=-1\) if \(u(t,x)\leq \alpha\), while \(h[u](t,x)=f(u(t,x))=1\) if \(u(t,x)\geq \beta\). When \(u(t,x)\in (\alpha,\beta)\), we set \(h[u](t,x)=f(u(\hat{t}(x),x))\), where \(\hat{t}(x)\leq t\) is the largest time in \([0,t]\) such that \(u(\hat{t}(x),x)\in\mathbb{R}\setminus (\alpha,\beta)\).
reviewed by: Philippe Laurençot
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 34C55
zbMATH DE Number: 6892223
zbMATH Keywords: discontinuous nonlinearity
zbMATH Keywords: interface
zbMATH Keywords: monotonicity formula
author: Darya E. Apushkinskaya
author: Nina Uraltseva
MaRDI profile type: Publication
cites work: Self-Similar Blow-Up and Hausdorff Dimension Estimates for a Class of Parabolic Free Boundary Problems
cites work: Free boundaries in problems with hysteresis
cites work: Control of self-organizing nonlinear systems
cites work: On regularity properties of solutions to the hysteresis-type problem
full work available at URL: s1064562418010167
OpenAlex ID: W2789604527
description: scientific article; zbMATH DE number 6892223