Related papers: Finite-dimensional global attractor for a nonlocal…
We study well-posedness and long-time dynamics of a class of quasilinear wave equations with a strong damping. We accept the Kirchhoff hypotheses and assume that the stiffness and damping coefficients are $C^1$ functions of the $L_2$-norm…
We investigate a cosmological model in which dark energy, represented by a quintessential scalar field, is coupled to a dark-matter perfect fluid in the spatially flat Friedmann-Robertson-Walker Universe. This allows an energy exchange in…
We obtain the analytical solutions to the Schr\"odinger equation for the attractive inverse-square potential in an induced electric dipole moment system under the influence of the harmonic oscillator. We show that bound states can exist…
We consider a system of several nonlinear equations with a distributed delay and obtain absolute asymptotic stability conditions, independent of the delay. The ideas of the proofs are based on the notion of a strong attractor. The results…
We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown…
This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions,…
Guided by classical concepts, we define the notion of \emph{ends} of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points…
The theory of non-gravitational interaction between a pressure-less dark matter (DM) and dark energy (DE) is a phenomenologically rich cosmological domain which has received magnificent attention in the community. In the present article we…
We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear"…
The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a…
The regularity of the global attractor of the incompressible Navier-Stokes equations for flows on two-dimensional domains is considered. It is assumed that domain $\Omega$ is a channel-like domain, i.e an arbitrary bounded or unbounded…
We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in…
We consider countable system of harmonic oscillators on the real line with quadratic interaction potential with finite support and local external force (stationary stochastic process) acting only on one fixed particle. In the case of…
We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial (PDE) and an ordinary differential equation (ODE), where both equations are coupled and…
We consider N run and tumble particles in one dimension interacting via a linear 1D Coulomb potential, an active version of the rank diffusion problem. It was solved previously for N = 2 leading to a stationary bound state in the attractive…
We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture…
A long-time behavior of solutions to a nonlinear plate model subject to non-conservative and non-dissipative effects and nonlinear damping is considered. The model under study is a prototype for a suspension bridge under the effects of…
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a…
This paper deals with the stabilization of a coupled system composed by an infinite-dimensional system and an ODE. Moreover, the control, which appears in the dynamics of the ODE, is subject to a general class of nonlinearities. Such a…
The aim of this paper is to study the finite-dimensional approximations of the nonautonomous lattice dynamical systems of the form $u_{i}'=\nu (u_{i-1}-2u_i+u_{i+1})-\lambda u_{i}+F(u_i)+f_{i}(t)\ (i\in \mathbb Z)\ (*)$. We show that the…