Existence Of Uniform Attractor Research Articles

Uniform attractors of non-autonomous suspension bridge equations with memory

In this article, we investigate the long-time dynamical behavior of non-autonomous suspension bridge equations with memory and free boundary conditions. We first establish the well-posedness of the system by means of the maximal monotone operator theory. Secondly, the existence of uniformly bounded absorbing set is obtained. Finally, asymptotic compactness of the process is verified, and then the existence of uniform attractors is proved for non-autonomous suspension bridge equations with memory term. For more information see https://ejde.math.txstate.edu/Volumes/2024/16/abstr.html

Dimension Estimate of Uniform Attractor for a Model of High Intensity Focussed Ultrasound-Induced Thermotherapy.

High intensity focussed ultrasound (HIFU) has emerged as a novel therapeutic modality, for the treatment of various cancers, that is gaining significant traction in clinical oncology. It is a cancer therapy that avoids many of the associated negative side effects of other more well-established therapies (such as surgery, chemotherapy and radiotherapy) and does not lead to the longer recuperation times necessary in these cases. The increasing interest in HIFU from biomedical researchers and clinicians has led to the development of a number of mathematical models to capture the effects of HIFU energy deposition in biological tissue. In this paper, we study the simplest such model that has been utilized by researchers to study temperature evolution under HIFU therapy. Although the model poses significant theoretical challenges, in earlier work, we were able to establish existence and uniqueness of solutions to this system of PDEs (see Efendiev et al. Adv Appl Math Sci 29(1):231-246, 2020). In the current work, we take the next natural step of studying the long-time dynamics of solutions to this model, in the case where the external forcing is quasi-periodic. In this case, we are able to prove the existence of uniform attractors to the corresponding evolutionary processes generated by our model and to estimate the Hausdorff dimension of the attractors, in terms of the physical parameters of the system.

Uniform attractors for measure-driven quintic wave equations

This is a detailed study of damped quintic wave equations with non-regular and non-autonomous external forces which are measures in time. In the 3D case with periodic boundary conditions, uniform energy-to- Strichartz estimates are established for the solutions, the existence of uniform attractors in a weak or strong topology in the energy phase space is proved, and their additional regularity is studied along with the possibility of representing them as the union of all complete bounded trajectories. Bibliography: 45 titles.

Attractors of multivalued semi-flows generated by solutions of optimal control problems

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.

Uniform Attractors for a Non-Autonomous Thermoviscoelastic Equation with Strong Damping

This paper considers the existence of uniform attractors for a non-autonomous thermoviscoelastic equation with strong damping in a bounded domain Ω⊆Rn(n≥1) by establishing the uniformly asymptotic compactness of the semi-process generated by the global solutions.

Uniform attractors for non-autonomous suspension bridge-type equations

We discuss the long-time dynamical behavior of the non-autonomous suspension bridge-type equation, where the nonlinearity is translation compact and the time-dependent external forces only satisfy Condition ( ) instead of being translation compact. By applying some new results and the energy estimate technique, the existence of uniform attractors is obtained. The result improves and extends some known results. MSC:34Q35, 35B40, 35B41.

Uniform Attractors for Non-Autonomous Strongly Damped Equations

In this paper, the vibration of beam equations with strongly damped term when the driving force is time-dependent is discussed. Based on the semigroup theorem and the technique of decomposing solutions, we prove the existence of compact kernel sections for the solution process. And under the homogeneous boundary condition, we obtain the existence of uniform attractors in a certain two-parameter region.

Dynamical behaviour of non-autonomous 2D Navier–Stokes equations with singularly oscillating external force

The purpose of this article is to analyse the dynamical behaviour of solutions of the non-autonomous 2D Navier–Stokes equations with singularly oscillating external force of the form: . First, we prove the existence of uniform attractor in for equations with external force non-translation compact. Then we prove that if the function g 1(z, t) satisfies the Divergence condition (Definition 3.1), then the uniform attractor is convergent to uniform attractor of the averaged equations in the sense of Hausdorff distance in L 2(Ω).

Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems

The existence of uniform attractor in is proved for the partly dissipative nonautonomous lattice systems with a new class of external terms belonging to , which are locally asymptotic smallness and translation bounded but not translation compact in . It is also showed that the family of processes corresponding to nonautonomous lattice systems with external terms belonging to weak topological space possesses uniform attractor, which is identified with the original one. The upper semicontinuity of uniform attractor is also studied.

Uniform attractors of non-autonomous dissipative semilinear wave equations*

Abstract The asymptotic long time behaviors of a certain type of non-autonomous dissipative semilinear wave equations are studied. The existence of uniform attractors is proved and their upper bounds for both Hausdorff and Fractal dimensions of uniform are given when the external force satisfies suitable conditions.

Attractors of nonautonomous schrödinger equations

The long-time behaviour of a two-dimensional nonautonomous nonlinear Schrodinger equation is considered. The existence of uniform attractor is proved and the upper bound of the uniform attractor's Hausdorff dimension is given.

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

The subject of this paper is the asymptotic behavior of a class of nonautonomous, infinite-dimensional dynamical systems with an underlying unbounded domain. We present an approach that is able to overcome both the law of compactness of the trajectories and the continuity of the spectrum of the linear part of the equations under consideration, providing nevertheless existence of uniform attractors. Moreover, our approach allows us to estimate the Hausdorff dimension of attractors of nonautonomous equations in terms of physical parameters. c 2000 John Wiley & Sons, Inc.