Related papers: Sharp bounds on the attractor dimensions for dampe…
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $\alpha>0$ and Ekman damping coefficient $\gamma>0$ is studied. We present explicit upper bounds for…
We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal…
In this paper we study the fractal dimension of global attractors for a class of wave equations with (single-point) degenerate nonlocal damping. Both the equation and its linearization degenerate into linear wave equations at the degenerate…
We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying…
We study the dimensions of the attractors for the fractional Navier--Stokes--Voigt equations. These equations, which include a fractional order of the Stokes operator applied to the time derivative, serve as natural extensions and…
We prove existence of the global attractor of the damped and driven Euler--Bardina equations on the 2D sphere and on arbitrary domains on the sphere and give explicit estimates of its fractal dimension in terms of the physical parameters.
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+\alpha u_t+\beta(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\times\Omega, u = 0,…
In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for…
We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter $\alpha>0$ and Ekman damping coefficient $\gamma>0$ endowed with periodic boundary conditions as well as their damped Euler limit…
Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-\Delta u+u+f(u)=0 \] in a bounded domain $\Omega$ in $\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term "massless…
In this paper, we study the initial boundary value problem for the two dimensional strong damped wave equation with exponentially growing source and damping terms. We first show the well-posedness of this problem and then prove the…
Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the…
The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of…
Consideration of various hydrodynamic phenomena involves the study of the Navier-Stokes (N-S) equations, what is hard enough for analytical and numerical investigations since already in three-dimensional (3D) case it is a challenging task…
In this paper, we study the longtime dynamics for the weakly damped wave equation with quintic non-linearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we establish the…
The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-\Delta)^\alpha\partial_t u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to…
There are presented examples of the rather sudden and violent explosion of the strange attractor of a one-dimensional driven damped anharmonic oscillator induced by a relatively small change of the amplitude of the strongly nonperturbative…
The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping exponent depends explicitly on time and may change sign. It is shown that in the case when the…
This paper is concerned with the initial boundary value problem for one dimensional strongly damped wave equation involving $p$-Laplacian. For $p>2$, we establish the existence of weak local attractors for this problem in…
The purpose of this paper is to investigate the existence and estimation of Hausdorff and fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain cause the…