Estimates on the dimension of an exponential attractor for a delay differential equation

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Sakineh Habibi
Keyword(s):  
Differential Equation ◽  
Fractal Dimension ◽  
Bounded Domain ◽  
Delay Differential Equation ◽  
Exponential Attractor ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Trajectory Method ◽  
Delay Differential

AbstractWe study the long time behavior of delay differential equation, considered in a bounded domain in ℝd. Using the short trajectory method to prove the existence of the exponential attractor. Also we have estimates on the fractal dimension of an exponential attractor.

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 Well-posedness and long-time behavior for the modified phase-field crystal equation

2014 ◽  
Vol 24 (14) ◽  
pp. 2743-2783 ◽  
Author(s):  
Maurizio Grasselli ◽  
Hao Wu
Keyword(s):  
Phase Field ◽  
Time Derivative ◽  
Exponential Attractor ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Global Existence And Uniqueness ◽  
And Diffusion ◽  
Phase Field Crystal

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K. R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the global existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the long-time behavior of the corresponding dynamical system. In particular, we establish the existence of the global attractor as well as an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space converges to a single equilibrium. This is done by means of a suitable version of the Łojasiewicz–Simon inequality. An estimate on the convergence rate is also given.

scholarly journals Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Junwei Feng ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Dynamical System ◽  
Bounded Domain ◽  
Multiplicative Noise ◽  
Random Dynamical System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Long Time ◽  
Uniform Attractors

<p style='text-indent:20px;'>In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.</p>

scholarly journals Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

2021 ◽  
Vol 7 (4) ◽  
pp. 4946-4959
Author(s):  
Ishtiaq Ali ◽  
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Higher Order ◽  
Proportional Delay ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Delay Differential ◽  
The Given ◽  
Past State

<abstract> <p>Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L<sub>∞</sub> norm is provided. Several numerical experiments were performed to confirm the theoretical results.</p> </abstract>

Sign in / Sign up

Export Citation Format

Share Document