Properties of Global Attractors of Partial Differential Equations

1992 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Global Attractors ◽  
Partial Differential

Exponential global attractors for semigroups in metric spaces with applications to differential equations

2011 ◽  
Vol 31 (6) ◽  
pp. 1641-1667 ◽  
Author(s):  
ALEXANDRE N. CARVALHO ◽  
JAN W. CHOLEWA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Global Attractor ◽  
Metric Space ◽  
Metric Spaces ◽  
Global Attractors ◽  
Invariant Sets ◽  
Bounded Subset ◽  
Partial Differential ◽  
Unstable Manifolds

AbstractIn this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

Attractors of Hamiltonian Nonlinear Partial Differential Equations

2021 ◽  
Author(s):  
Alexander Komech ◽  
Elena Kopylova
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Global Attractors ◽  
Stationary States ◽  
Partial Differential ◽  
Open Problems ◽  
Novel Applications ◽  
Mathematics And Physics

This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.

Analyticity for a class of non-linear evolutionary pseudo-differential equations

2014 ◽  
Vol 25 (6) ◽  
pp. 783-793 ◽  
Author(s):  
XENAKIS IOAKIM ◽  
YIORGOS-SOKRATIS SMYRLIS
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spectral Method ◽  
Wide Class ◽  
Global Attractors ◽  
Periodic Functions ◽  
Partial Differential ◽  
Suitable Norm ◽  
Non Linear ◽  
Spatially Periodic

We study the analyticity properties of solutions for a class of non-linear evolutionary pseudo-differential equations possessing global attractors. In order to do this we utilise an analyticity criterion for spatially periodic functions, which involves the rate of growth of a suitable norm of the nth derivative of the solution, with respect to the spatial variable, as n tends to infinity. This criterion can be used to a wide class of dissipative-dispersive partial differential equations, provided they possess global attractors. Using this criterion and the spectral method developed in Akrivis et al. [1] we have improved previous results.

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