Longtime behavior of semilinear reaction-diffusion equations on the whole space
type: Item
instance of: scholarly article
title: Longtime behavior of semilinear reaction-diffusion equations on the whole space
zbMATH Open document ID: 1030.35102
author: Vittorino Pata
author: Claudio Santina
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2004-02-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108552
review text: The paper deals with the following semilinear reaction-diffusion equation on the whole space \[ u_t - \Delta u + g(x,u) = f , \;\;(x,t) \in \mathbb{R}^3 \times (\tau, T],\qquad u(x,\tau)= u_0(x) \] where \(u= u(x,t)\), \(f=f(x,t)\). The authors prove existence, uniqueness and continuous dependence results. Moreover, they show the existence of absorbing sets and find, in the autonomous case, a universal attractor for the associated semigroup. All the stated results are also valid for systems in \(\mathbb{R}^n\), that is when \(u\), \(f\), \(g\) are \(m\)-dimensional vectors and \(x \in \mathbb{R}^n\). The authors develop a specific technique, in order to overcome the difficulty caused by the lack of compactness of the usual Sobolev embeddings in unbounded domains.
reviewed by: Cristina Marcelli
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 2035106
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2035106