Narrow Results

The existence of a pullback attractor in L²(Ω) for the following non-autonomous reaction–diffusion equation

In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

In this paper, we prove the existence of pullback and uniform attractors for a nonautonomous Liénard equation. The relation among these attractors is also discussed. After that, we consider that the Liénard equation includes forcing terms which belong to a class of functions extending periodic and almost periodic functions recently introduced by Kloeden and Rodrigues [2011]. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate these results with a numerical simulation: we present a simulation showing the pullback attractor for the nonautonomous Van der Pol equation, an important special case of the nonautonomous Liénard equation.

The existence of minimal pullback attractors in $L^2(\mathrm{\Omega })$ for a nonautonomous reaction–diffusion equation, in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition, is proved in this paper, when the domain $\mathrm{\Omega }$ is a general nonempty open subset of ${ℝ}^N$, and $h\in L_{\mathrm{\text{loc}}}^2(ℝ;H^{-1}(\mathrm{\Omega }))$. The main concept used in the proof is the asymptotic compactness of the process generated by the problem. The relation among these families is also discussed.

The existence of a pullback attractor for the nonautonomous $p$-Laplacian type equations on infinite lattices is established under certain natural dissipative conditions. In particular, there is no restriction on the power index $q$ of the nonlinearity relative to the index $p$. The forward limiting behavior is also discussed and, under suitable assumptions on the time dependent terms, the lattice system is shown to be asymptotically autonomous with its pullback attractor component sets converging upper semi-continuously to the autonomous global attractor of the limiting autonomous system.

This paper investigates the pullback attractor of Cohen–Grossberg neural networks with multiple time-varying delays. Compared with the existing references, the networks considered here are more general and cannot be expressed in the vector-matrix form due to multiple time-varying delays. After constructing a proper Lyapunov–Krasovskii functional and eliminating the terms involving multiple time-varying delays, two sets of new sufficient criteria on the existence of the pullback attractor are derived based on the theory of pullback attractors. In the end, two examples are given to demonstrate the effectiveness of our theoretical results.

This is a systematic study of global pullback attractors of -analytic cocycles. For the large class of -analytic cocycles we give the description of structure of their pullback attractors. In particular we prove that it is trivial, i.e. the fibers of these attractors contain only one point. Several applications of these results are given (ODEs, Caratheodory's equations with almost periodic coefficients, almost periodic ODEs with impulse).

We consider the two-step bifurcation scenario which has been studied by L. Arnold and his co-workers. We formulate a "continuous case" and a "measurable case" of the scenario, and present results and conjectures regarding sufficient conditions that it take place.

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

We prove the existence and uniqueness of tempered random attractors for stochastic Reaction–Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the tempered attractors when the stochastic equations are forced by periodic functions. We further prove the upper semicontinuity of these attractors when the intensity of stochastic perturbations approaches zero.

Using a method to prove the pullback asymptotically compactness for the multi-valued processes, we present the existence of unique pullback attractors in C_V,H and C_D(A),V for the multi-valued process associated with a damped wave equation with delays and without the uniqueness of solutions.

This paper is concerned with the dynamical behavior of the damped Boussinesq equation containing a delay forcing term. It shows that, under suitable dissipative conditions, such a system possesses a pullback attractor and a uniform forward attractor in ${{𝒞}}_{X_0}$.

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to prove the existence of pullback attractors for the multi-valued non-compact random dynamical systems associated with the semi-linear degenerate parabolic unbounded delay equations with both deterministic and random external terms.

In this paper, we prove the existence of a pullback attractor for a strongly damped delay wave equation in ${ℝ}^n$. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions, so that some new methods to obtain the existence of pullback attractors for multi-valued processes on an unbounded domain are introduced.

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction–diffusion equation with a nonlinear multiplicative noise. The existence and uniqueness of solutions and weak pullback mean random attractors are also established for a deterministic non-local reaction–diffusion equations with random initial data.

Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.