Exponential Attractors in Strong Topology Space for a Nonclassical Diffusion Equation

2013 ◽  
Vol 15 (3) ◽  
pp. 285
Author(s):  
Zhuanling LIU
Keyword(s):  
Diffusion Equation ◽  
Strong Topology ◽  
Exponential Attractors ◽  
Nonclassical Diffusion Equation

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

Global Attractors for a Nonclassical Diffusion Equation

2007 ◽  
Vol 23 (7) ◽  
pp. 1271-1280 ◽  
Author(s):  
Chun You Sun ◽  
Su Yun Wang ◽  
Cheng Kui Zhong
Keyword(s):  
Diffusion Equation ◽  
Global Attractors ◽  
Nonclassical Diffusion Equation

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 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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