# Cone Invariance Properties

• P. Constantin
• C. Foias
• B. Nicolaenko
• R. Teman
Part of the Applied Mathematical Sciences book series (AMS, volume 70)

## Abstract

One of the features of a dissipative equation of type (2.1), (2.2) is the existence of compact absorbing sets. More precisely, there exists YH satisfying
$$Y{\text{ is convex, closed in }}H,{\text{ a bounded neighborhood or 0 in }}\mathcal{D}{\text{ (}}{A^{1/4}}{\text{) (in particular, }}Y{\text{ is compact in }}H).{\text{ }}$$
(5.1)
$${\text{For every }}\theta {\text{ }} \geqslant {\text{ 1 and any }}{u_o} \in \theta Y{\text{ the inequalities (3}}{\text{.7), (3}}{\text{.8), and (4}}{\text{.5a) are valid}}{\text{. The constants }}{k_{\text{1}}}{\text{, }}{k_{\text{2}}}{\text{, }}{k_{\text{3}}}{\text{, }}{k_{\text{5}}}{\text{, and }}{k_{\text{6}}}{\text{ depend on }}\theta {\text{ only}}{\text{. }}$$
(5.2)
$${\text{The set }}Y{\text{ is absorbing; i}}{\text{.e}}{\text{., if }}Z{\text{ is any bounded set in }}H{\text{, there exists a }}{{\text{t}}_{\text{o}}}{\text{ }} \geqslant {\text{ 0 (depending on }}Z{\text{) such that }}S{\text{(}}t{\text{)}}Z{\text{ }} \subset {\text{ }}Y{\text{ for }}t{\text{ }} \geqslant {\text{ }}{t_{\text{o}}}{\text{.}}$$
(5.3)

## Keywords

Differential Equation Partial Differential Equation Initial Data Simple Computation Precise Statement
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1989

## Authors and Affiliations

• P. Constantin
• 1
• C. Foias
• 2
• B. Nicolaenko
• 3
• R. Teman
• 4
1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
3. 3.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
4. 4.Department de MathematiquesUniversité de Paris-SudOrsayFrance