A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

2020 ◽  
Vol 44 (1) ◽  
pp. 91-103 ◽  
Author(s):  
Brahim Alouini
Keyword(s):  
Fractal Dimension ◽  
Global Attractors ◽  
Nonlinear Schrödinger ◽  
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.

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.

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

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.

scholarly journals Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

2021 ◽  
Vol 6 (11) ◽  
pp. 11778-11795
Author(s):  
Jianbo Yuan ◽  
◽  
Shixuan Zhang ◽  
Yongqin Xie ◽  
Jiangwei Zhang ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Polynomial Growth ◽  
Arbitrary Order ◽  
Dynamical Behavior ◽  
Global Attractors ◽  
Diffusion Equations ◽  
Asymptotic Regularity ◽  
Exponential Attractors ◽  
Nonclassical Diffusion Equation ◽  
Nonclassical Diffusion Equations

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

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