Global attractor for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we study the existence of global attractors for the extensible beam equation with localized nonlinear damping and linear memory. Copyright © 2011 John Wiley & Sons, Ltd.     

Weak solutions for the Falk model system of shape memory alloys in energy class

We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Global attractor for a density-dependent sensitivity chemotaxis model

In this paper, a chemotaxi model with reproduction term in a bounded domain Ω ⊂ Rⁿ is discussed. The existence of a global-in-time solution and a global attractor for this model are obtained.     

Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions

This paper is concerned with the following one-dimensional nonlinear system of equations:

(0.1)     

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.     

Global attractor for the semilinear thermoelastic problem

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Global existence to a three‐dimensional non‐linear thermoelasticity system arising in shape memory materials

This paper is concerned with the unique global solvability of a three‐dimensional (3‐D) non‐linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non‐convex elastic energy and non‐linear heat conduction with mechanical dissipation. The present paper extends the previous 2‐D existence result of the authors Reference [1] to 3‐D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.     

Global attractor in autonomous competitive Lotka-Volterra systems

For autonomous Lotka-Volterra systems modelling the dynamics of competing species, a new condition has been found to prevent a particular species from dying out. Based on this condition, criteria have been established for all or some of the species to stabilise at a steady state whilst the others, if any, die out.

     

Global attractor for heat convection problem in a micropolar fluid

The article is devoted to describe asymptotics in the heat convection problem for a micropolar fluid in two dimensions. We show the existence and the uniqueness of global in time solutions and then prove the existence of a global attractor for considered model. Next, the Hausdorff dimension of the global attractor is estimated. Copyright © 2006 John Wiley & Sons, Ltd.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

具有非线性记忆项的耗散波动方程的整体吸引子

LetΩRn be a bounded domain with a smooth boundary.We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term utt+αut-△u-∫0t 0μ(t-s)|u(s)| βu(s)ds + g(u)=f.Based on a time-uniform priori estimate method,the existence of the compact global attractor is proved for this model in the phase space H10(Ω)×L2(Ω).     

Global attractor in competitive Lotka-Volterra systems with retardation

Global attractivity is studied for a class of competitive Lotka-Volterra differential systems with retardation. Sufficient conditions, which contain a number of existing results as special instances, are provided for a system to have a single-point global attractor. By these conditions, predictions can be made either for coexistence and stability of all the species or for balance of survival and extinction.     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

Control problem for a non‐linear thermoelasticity system

The control problem for a three‐dimensional non‐linear thermoelasticity system is considered. The system may represent, among others, the dynamical model of shape memory materials. As controls we take distributed heat sources and body forces. The goal functional refers to the desired evolution of displacement, strain and temperature. The continuity and differentiability of solutions with respect to controls is studied. The existence of optimal controls is proved and the necessary optimality conditions are formulated. The existence of adjoint state variables is proved under additional regularity of data. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of the steady state for the Falk model system of shape memory alloys

In this article a stability result for the Falk model system is proven. The Falk model system describes the martensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal elliptic problem. We show the dynamical stability for the linearized stable critical point of the corresponding functional. Copyright © 2007 John Wiley & Sons, Ltd.     

Global attractor in competitive Lotka–Volterra systems

For autonomous Lotka–Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.     

Global attractor for a low order ODE model problem for transition to turbulence

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy‐conserving, non‐quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.