Homogenization of monotone parabolic problems with several temporal scales.
type: Item
instance of: scholarly article
title: Homogenization of monotone parabolic problems with several temporal scales.
zbMATH Open document ID: 1265.35018
author: Jens Persson
published in: Applications of Mathematics
publication date: 2013-01-02T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/142850
review text: The author studies the asymptotic behaviour, as \(\varepsilon \) goes to \(0\), of the sequence of weak solutions \(u_\varepsilon \) in \(H^1(0,T;H^1_0(\Omega ),H^{-1}(\Omega ))\) of monotone parabolic problems of the type \[ \frac {\partial }{\partial t} u_\varepsilon (x,t) - \nabla \cdot a\left ( x,t,\frac {x}{\varepsilon },\frac {t}{\varepsilon ^{\prime }_1}, \ldots , \frac {t}{\varepsilon ^{\prime }_m};\nabla u_\varepsilon (x,t)\right )=f(x,t)\; \; \text{ in } \Omega \times (0,T) \] \(u_{\varepsilon }(x,0) = u_0(x)\) in \(\Omega \), \(u_{\varepsilon }(x,t)=0\) on \(\partial \Omega \times (0,T)\), where \(\Omega \subset \mathbb {R}^N\) is an open bounded set with the Lipschitz boundary, \(T>0\), and there exist two well-separated spatial scales \(\{1, \varepsilon \}\), the macroscopic and the microscopic, and \((m+1)\) well-separated temporal scale functions \(\{1,\varepsilon ^\prime _1,\ldots , \varepsilon ^{\prime }_m\}\). Assuming that the function \(a\:\Omega \times (0,T)\times \mathbb {R}^N\times \mathbb {R}^m\times \mathbb {R}^N\to \mathbb {R}^N\) satisfies additional continuity, periodicity and growth conditions, it is proved that the sequence \(\{u_{\varepsilon }\}\) weak converges to \(u\) in \(L^2(0,T;H^1_0(\Omega ))\), and the sequence \(\{\nabla u_\varepsilon \)\} \((2,m+1)\)-scale converge to \(\nabla u+\nabla _y u_1\), where \(u\) is the unique solution of the homogenized problem \[ \frac {\partial }{\partial t} u_ (x,t) - \nabla \cdot b\left ( x,t, ;\nabla u (x,t)\right ) = f(x,t)\; \; \text{ in } \Omega \times (0,T), \] \(u (x,0)=u_0(x)\) in \(\Omega \), \(u (x,t)=0\) on \(\partial \Omega \times (0,T)\) and the characterization of \(u_1\) is given in the following cases: the temporal oscillations are slower than the spatial one, the slow spatial and temporal oscillations are in resonance, the temporal oscillations are faster than the spatial one, the rapid spatial and temporal oscillations are in resonance.
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 6121189
zbMATH Keywords: homogenization
zbMATH Keywords: \(H\)-convergence
zbMATH Keywords: multiscale convergence
zbMATH Keywords: parabolic problem
MaRDI profile type: Publication
OpenAlex ID: W2051615927
cites work: Homogenization and Two-Scale Convergence
cites work: Multiscale convergence and reiterated homogenisation
cites work: Q4189539
cites work: The perturbed test function method for viscosity solutions of nonlinear PDE
cites work: Q5455267
cites work: Homogenization of some parabolic operators with several time scales.
cites work: Reiterated homogenization of monotone parabolic problems
cites work: Homogenization of parabolic equations---an alternative approach and some corrector-type results
cites work: Multiscale convergence and reiterated homogenization of parabolic problems.
cites work: Reiterated homogenization of nonlinear monotone operators
cites work: SCALE CONVERGENCE IN HOMOGENIZATION
cites work: A General Convergence Result for a Functional Related to the Theory of Homogenization
cites work: Homogenization structures and applications. I
cites work: Q4816721
cites work: \(\Sigma\) -convergence of nonlinear parabolic operators
cites work: Q5534497
cites work: Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales
DOI: 10.1007/S10492-012-0013-Z
description: scientific article; zbMATH DE number 6121189