Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

Displacement potentials for functionally graded piezoelectric solids

Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.     

Global attractors and approximate inertial

An abstract nonautonomous differential equation, u' + Au + F(u) = f ( t ) , is considered using assumptions appropriate for systems of reaction-diffusion equations on multi-dimensional spatial domains. A priori estimates establish the existence of absorbing balls in relevant function spaces, and nonsequently the existence of a global attractor is verified in the associated skew-product flow. It is also shown that approximate inertial manifolds exist for this equation. These are finite-dimensional manifolds which have exponentially attracting neighborhoods under the flow. The dimension of the manifold and the thickness of the attracting neighborhood are inversely related.     

Global attractors for gradient flows in metric spaces

We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope of Ambrosio et al. (2005) [5], we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach (De Giorgi, 1993; Ambrosio, 1995 [3], [15]).For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced in Ball (1997) [7].The notions of generalized and energy solutions are quite flexible, and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces, to Wasserstein spaces of probability measures. We present applications of our abstract results, by proving the existence of the global attractor for the energy solutions, both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.     

Global attractors for the extensible thermoelastic beam system

This work is focused on the dissipative system

     

Global random attractors for the stochastic dissipative Zakharov equations

The stochastic dissipative Zakharov equations with white noise are mainly investigated. The global random attractors endowed with usual topology for the stochastic dissipative Zakharov equations are obtained in the sense of usual norm. The method is to transform the stochastic equations into the corresponding partial differential equations with random coefficients by Ornstein-Uhlenbeck process. The crucial compactness of the global random attractors wiil be obtained by decomposition of solutions.     

Global attractors for porous-elasticity system from second spectrum viewpoint

This paper is concerned with the global attractors for a new partially damped porous-elasticity system by taking a truncated version which is free of blow-up on second wave speed. We establish the global well-posedness of the system via Faedo–Galerkin method. By considering only one damping term acting on the volume fraction in the system we prove the existence of absorbing set for the solution semigroup regardless any relationship between coefficients of the system. Finally, by using Lyapunov and recent quasi-stability methods we prove the existence of smooth global attractors with finite fractal dimension.     

Global attractors for the wave equation with nonlinear damping

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R³ with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C₁(|s|−C₂)?|g(s)|?C₃(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f^′(s)|?C₄(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].     

Global attractors for singular perturbations of the Cahn-Hilliard equations

We consider the singular perturbations of two boundary value problems, concerning respectively the viscous and the nonviscous Cahn-Hilliard equations in one dimension of space. We show that the dynamical systems generated by these two problems admit global attractors in the phase space , and that these global attractors are at least upper-semicontinuous with respect to the vanishing of the perturbation parameter.     

Global attractors for quasilinear parabolic equations on unbounded thin domains

Monatshefte für Mathematik - In this paper we are concerned with the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower...     

Global attractors for von Karman equations with nonlinear interior dissipation

In this paper we study the asymptotic behavior of weak solutions for von Karman equations with nonlinear interior dissipation. We prove the existence of a global attractor in the space .     

Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients.     

Global attractors for the coupled suspension bridge system with temperature

This paper deals with the long‐term properties of the thermoelastic nonlinear string‐beam system related to the well‐known Lazer–McKenna suspension bridge model (0.1) In particular, no mechanical dissipation occurs in the equations, because the loss of energy is entirely due to thermal effects. The existence of regular global attractors for the associated solution semigroup is proved (without resorting to a bootstrap argument) for time‐independent supplies f,g,h and any . Copyright © 2015 John Wiley & Sons, Ltd.     

Global attractors for a discrete selection model with periodic immigration

A one-island selection-migration model is used to study the periodic immigration of a population of fixed allele frequency into a natural population. Density-dependent selection and immigration are the primary factors affecting the demographic and genetic change in the island population. With the assumptions of complete dominance (CD) or no dominance (ND) and homozygote superiority in fitness, the existence and location of global attractors are established. Analysis of this model provides rudimentary information about the migration of transgenes into a natural population.     

Global attractors for nonlinear wave equations with nonlinear dissipative terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x,ut)=a(x)r|ut|ut.     

Global and exponential attractors for a Caginalp type phase-field problem

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.