Pullback Exponential Attractors for Nonautonomous Reaction–Diffusion Equations

2015 ◽  
Vol 25 (05) ◽  
pp. 1550063
Author(s):  
Xingjie Yan ◽  
Wei Qi
Keyword(s):  
A Priori ◽  
Reaction Diffusion ◽  
A Priori Estimate ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Estimate Method ◽  
Necessary And Sufficient ◽  
Asymptotic A Priori Estimate

This paper presents a necessary and sufficient condition to prove the existence of the pullback exponential attractor. The asymptotic a priori estimate method is used to produce an abstract result on the existence of the pullback exponential attractor in a strong space without regularity. The established results are illustrated by applying them to the nonautonomous reaction–diffusion equations to prove the existence of the pullback exponential attractors in L2(Ω), [Formula: see text] and Lp(Ω)(p > 2) spaces.

scholarly journals PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMI-LINEAR DEGENERATE PARABOLIC EQUATION

2010 ◽  
Vol 52 (3) ◽  
pp. 537-554 ◽  
Author(s):  
CUNG THE ANH ◽  
TANG QUOC BAO
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
A Priori ◽  
Reaction Diffusion ◽  
A Priori Estimate ◽  
Reaction Diffusion Equations ◽  
Pullback Attractors ◽  
Estimate Method ◽  
Degenerate Parabolic ◽  
Linear Degenerate

AbstractIn this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.

scholarly journals Nonnegative solutions to the reaction-diffusion equations for prey-predator models with the dormancy of predators

2021 ◽  
Author(s):  
Novrianti Novrianti ◽  
Okihiro Sawada ◽  
Naoki Tsuge
Keyword(s):  
A Priori ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Classical Solutions ◽  
Evolution Operators ◽  
A Priori Bounds ◽  
Nonnegative Solutions ◽  
Invariant Regions ◽  
Global Unique Solvability

The time-global unique solvability on the reaction–diffusion equations for preypredator models and dormancy on predators is established. The crucial step is to construct time-local nonnegative classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principles, a priori bounds are derived, so solutions are extended time-globally. Via observations to the corresponding ordinary differential equations, invariant regions and asymptotic behaviors of solutions are also investigated.

scholarly journals Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoming Fan
Keyword(s):  
Dynamical System ◽  
Fractal Dimension ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Exponential Attractor ◽  
Spatial Discretization ◽  
First Order ◽  
Lattice Dynamical System

We construct an exponential attractor for a first-order dissipative lattice dynamical system arising from spatial discretization of reaction-diffusion equations in . And we obtain fractal dimension of the exponential attractor.

scholarly journals Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yongqin Xie ◽  
Jun Li ◽  
Kaixuan Zhu
Keyword(s):  
Polynomial Growth ◽  
A Priori ◽  
Upper Semicontinuity ◽  
A Priori Estimate ◽  
Global Attractors ◽  
Diffusion Equations ◽  
Nonlinear Anal ◽  
Estimate Method ◽  
Appl Math ◽  
Nonclassical Diffusion Equations

AbstractIn this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order $p-1$ p − 1 ($p \geq 2$ p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

2019 ◽  
Vol 84 (4) ◽  
pp. 669-678
Author(s):  
Lennon Ó Náraigh ◽  
Khang Ee Pang
Keyword(s):  
Steady State ◽  
A Priori ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Mathematical Framework ◽  
Source Terms ◽  
The Stability

Abstract We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction–diffusion equations with periodic source terms in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch’s theorem and Poincaré’s inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

scholarly journals A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 600 ◽  
Author(s):  
Jie Zhao ◽  
Hong Li ◽  
Zhichao Fang ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
A Priori ◽  
Reaction Diffusion ◽  
Auxiliary Variable ◽  
Volume Element ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Unknown Variable ◽  
Finite Volume Element ◽  
Fractional Reaction

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

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