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In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system. [ABSTRACT FROM AUTHOR]

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearity u t − div (D (x) | ∇ u | p (x) − 2 ∇ u) = f (x , t , u) − A (x) | u | q (x) − 2 u in the cylinder Ω × (0 , T) with given nonnegative weights D (x) , A (x) , measurable bounded exponents p (x) ∈ [ p − , p + ] , q (x) ∈ [ q − , q + ] and a globally Lipschitz function f (x , t , u). Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents p (x) , q (x) which guarantee that the associated semigroup has a compact global attractor in L 2 (Ω). It is shown that in case the exponents p (x) and q (x) do not meet the sufficient conditions of existence of a nontrivial global attractor and ‖ u (0) ‖ L 2 (Ω) is sufficiently small, then every solution with bounded ‖ u (t) ‖ L 2 (Ω) 2 either vanishes in a finite time, or decays exponentially as t → ∞. [ABSTRACT FROM AUTHOR]

Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077-1105], we considered the conserved phasefield system {τφt - Δ(δφt - Δφ + g(φ) - u) = 0, εut + φt - Δu = 0, in a bounded domain of Rd , d = 1, 2, 3, where τ > 0 is a relaxation time, δ > 0 is the viscosity parameter, ε ∈ (0, 1] is the heat capacity, φ is the order parameter, u is the absolute temperature and g : R → R is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at ε = 0. Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at (ε, δ) = (0, 0). To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε, we require ε to be dominated from above by δ. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn-Hilliard equation, improving and extending some previous results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the upper semicontinuity of global attractors for a damped wave equations with perturbations. [ABSTRACT FROM AUTHOR]

We consider a modified Cahn-Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αut, α≥0. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history space. Existence of a variational solution is obtained. Then, in the case α>0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case α=0, we only establish the existence of a trajectory attractor. [ABSTRACT FROM AUTHOR]

We consider a nonconserved phase-field model of Caginalp type which has been proposed by H.G. Rotstein et al. This model consists of two coupled integro-partial differential equations, one for the temperature ϑ, the other for the order parameter χ, and each one is characterized by a nonnegative decreasing memory kernel. Such equations, endowed with homogeneous Dirichlet (for ϑ) and Neumann (for χ) boundary conditions, generate a dissipative dynamical system which possesses a global attractor 𝒜. This fact was proven by V. Pata and the author. Here we construct a compact exponentially attracting set which, in particular, entails a smoothness result for 𝒜. Then we also demonstrate the convergence of any trajectory to a single equilibrium, via the Łojasiewicz–Simon approach. [ABSTRACT FROM AUTHOR]