A note on attractors with finite fractal dimension

2008 ◽  
Vol 40 (4) ◽  
pp. 651-658 ◽  
Author(s):  
Radoslaw Czaja ◽  
Messoud Efendiev
Keyword(s):  
Fractal Dimension ◽  
Finite Fractal Dimension

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

scholarly journals Fractal Dimension of a Random Invariant Set and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Gang Wang ◽  
Yanbin Tang
Keyword(s):  
Fractal Dimension ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Invariant Set ◽  
Invariant Sets ◽  
Abstract Result ◽  
Random Attractors ◽  
Finite Fractal Dimension ◽  
Degenerate Parabolic ◽  
Random Invariant Set

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.

Asymptotic dynamics of the solutions for a system of N-coupled fractional nonlinear Schrödinger equations

2020 ◽  
pp. 1-24
Author(s):  
Brahim Alouini
Keyword(s):  
Fractal Dimension ◽  
Global Attractor ◽  
Current Issue ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Schrodinger Equations ◽  
Finite Fractal Dimension ◽  
Asymptotic Dynamics

In the current issue, we consider a system of N-coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities. We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor with finite fractal dimension.

scholarly journals SEMIMARTINGALE ATTRACTORS FOR ALLEN–CAHN SPDEs DRIVEN BY SPACE–TIME WHITE NOISE I: EXISTENCE AND FINITE DIMENSIONAL ASYMPTOTIC BEHAVIOR

2004 ◽  
Vol 04 (02) ◽  
pp. 223-244 ◽  
Author(s):  
H. ALLOUBA ◽  
J. A. LANGA
Keyword(s):  
Asymptotic Behavior ◽  
Fractal Dimension ◽  
White Noise ◽  
Global Attractors ◽  
Space Time ◽  
Finite Dimensional ◽  
Determining Modes ◽  
Random Attractors ◽  
Finite Fractal Dimension ◽  
Regularity Results

We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa [5]. In this paper we focus on second-order SPDEs of the Allen–Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and for the corresponding random PDEs of Allen–Cahn type; we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite-dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of these random attractors and give a result on determining modes, both in the forward and the pullback senses.

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