scholarly journals Random Pullback Attractor of a Non-autonomous Local Modified Stochastic Swift-Hohenberg with Multiplicative Noise

2020 ◽  
Author(s):  
Yongjun Li ◽  
Tinggang Zhao ◽  
Hongqing Wu
Keyword(s):  
External Force ◽  
Exponential Growth ◽  
Multiplicative Noise ◽  
Pullback Attractor ◽  
Stochastic Term

In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.

scholarly journals PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS GENERALIZED CAHN-HILLIARD EQUATION WITH BIOLOGICAL APPLICATIONS

2016 ◽  
Vol 21 (3) ◽  
pp. 371-384
Author(s):  
Ning Duan
Keyword(s):  
External Force ◽  
Exponential Growth ◽  
Pullback Attractor ◽  
Biological Applications ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

In this paper, we consider a non-autonomous generalized Cahn-Hilliard equation with biological applications. It is shown that a pullback attractor of the equation exists when the external force has exponential growth.

scholarly journals The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yongjun Li ◽  
Xiaona Wei ◽  
Yanhong Zhang
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

scholarly journals Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nguyen Dinh Binh
Keyword(s):  
Exponential Growth ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equations ◽  
Pullback Attractor ◽  
Pullback Attractors ◽  
Grushin Operator ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domainΩ. We prove the regularity and exponential growth of a pullback attractor in the spaceS02(Ω)∩L2p−2(Ω)for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.

scholarly journals Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

2019 ◽  
Vol 17 (1) ◽  
pp. 1411-1434
Author(s):  
Zongfei Han ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Multiplicative Noise ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Random Dynamical System ◽  
Exponential Attractor

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

scholarly journals Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

2016 ◽  
Vol 14 (1) ◽  
pp. 586-602 ◽  
Author(s):  
Xiaoyao Jia ◽  
Juanjuan Gao ◽  
Xiaoquan Ding
Keyword(s):  
Dynamical System ◽  
Neumann Boundary Condition ◽  
Multiplicative Noise ◽  
Space Dimension ◽  
Stochastic Equation ◽  
Uniform Estimate ◽  
Neumann Boundary ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Homogeneous Neumann Boundary Condition

Abstract In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

scholarly journals Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yantao Guo ◽  
Shuilin Cheng ◽  
Yanbin Tang
Keyword(s):  
External Force ◽  
Distributed Delay ◽  
Navier Stokes ◽  
Pullback Attractor ◽  
Sufficient Condition ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Prior Estimate ◽  
Long Time ◽  
Voigt Fluid

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation.

scholarly journals Statistical Solutions for a Nonautonomous Modified Swift-Hohenberg Equation

2021 ◽  
Author(s):  
Jintao Wang ◽  
Xiaoqian Zhang ◽  
Caidi Zhao
Keyword(s):  
Sobolev Space ◽  
External Force ◽  
Smooth Domain ◽  
Probability Measures ◽  
Pullback Attractor ◽  
Statistical Solution ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Borel Probability ◽  
Statistical Solutions

We consider the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x)$$ on a bounded smooth domain $\Omega\subset\R^n$ with $n\leqslant 3$. We show that, if $|b|<4$ and the external force $g$ satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space $H_0^2(\Omega)$. Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

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