Long time behavior of solutions for a nonlinear evolution equation

2005 ◽  
Vol 63 (5-7) ◽  
pp. e1199-e1206 ◽  
Author(s):  
Kei Matsuura
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

Averaging of Attractors and Inertial Manifolds for Parabolic PDE With Random Coefficients

2005 ◽  
Vol 5 (4) ◽  
Author(s):  
Igor D. Chueshov ◽  
Björn Schmalfuß
Keyword(s):  
Evolution Equation ◽  
Time Scale ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Fast Time ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Scaling Parameter ◽  
Fast Time Scale ◽  
Long Time

AbstractThe averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

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.

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