Uniform long-time behavior of solutions of parabolic equations depending on slow time
type: Item
instance of: scholarly article
title: Uniform long-time behavior of solutions of parabolic equations depending on slow time
zbMATH Open document ID: 0915.35050
author: K. Appert
published in: Journal of Differential Equations
publication date: 1999-06-23T00:00:00Z
review text: The reaction-diffusion system \[ \partial_t=a\Delta u-f(u,\lambda t)-g_0, \quad u| _{\partial\Omega}=0 \] is considered with the initial condition \(u| _{t=\theta}=u_\theta\). Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary, \(u=(u^1(x,t),\dots,u^m(x,t))=u(x,t)\) is a vector-function, \(x\in\Omega\), \(t\geq 0\). \(a\) is a diagonal matrix \(a=\text{ diag }(\nu_1,\dots,\nu_m)\), \(\nu_i\geq\nu_0>0\). The function \(g_0=(g^1,\dots,g^m)\) depends only on \(x\). The nonlinearity \(f(u,\tau)=(f^1(u,\tau),\dots,f^m(u,\tau))\) is potential, that is \(f_i(u,\tau)=\partial F(u,\tau)/\partial u^i\), \(i=1,\dots,m\), where \(F(0,\tau)=0\). It is assumed that \(f\) is continuously differentiable on \(\mathbb{R}^m\times \mathbb{R}\) and that \(f(u,\tau)u\geq -C\). This system generates a semigroup \(\{S_t\}\) in the space \(E=(H_1(\Omega)\cap\{u| _{\partial\Omega}=0\})\times \mathbb{R}\times \mathbb{R}\). Local invariant manifolds of the semigroup \(\{ S_t\}\) and exponential attractors consisting of invariant manifolds are constructed.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 1247707
zbMATH Keywords: invariant manifolds
zbMATH Keywords: exponential attractors
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3450
OpenAlex ID: W2001794298
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DOI: 10.1006/JDEQ.1998.3450
description: scientific article; zbMATH DE number 1247707