The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

2020 ◽  
Author(s):  
Ming Wang ◽  
Jianhua Huang
Keyword(s):  
Fractal Dimension ◽  
Global Attractor ◽  
Kdv Equation ◽  
Finite 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.

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 Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

2019 ◽  
Vol 20 (2) ◽  
pp. 485-515
Author(s):  
Jan W. Cholewa ◽  
Radosław Czaja
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Global Attractor ◽  
American Mathematical Society ◽  
Reaction Diffusion Equations ◽  
Dissipative Systems ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Exponential Attractors ◽  
Lattice Dynamical Systems

Abstract In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in $$\ell ^2$$ℓ2, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

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.

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