Related papers: Sharp bounds on the attractor dimensions for dampe…
In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail…
We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems…
This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $\mathbb{R}^3$ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global…
We consider the semilinear reaction diffusion equation $\partial_t\phi-\nu\Delta\phi-V(x)\phi+f(\phi)=0$, $\nu>0$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$…
This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of $\mathbb{R}^3$ with fractional damping of the form $(-\Delta_x+1)^\frac{1}{2}\partial_t u$. The work extends previously known…
For a damped wave (or Klein-Gordon) equation on a bounded domain, with a focusing power-like nonlinearity satisfying some growth conditions, we prove that a global solution is bounded in the energy space, uniformly in time. Our result…
In this paper, we apply an implicit Euler scheme to discretize the complex Ginzburg-Landau equation and prove the existence of a numerical attractor for the discrete Ginzburg-Landau system. We establish the upper semicontinuity of the…
We consider a class of wave equations with constant damping and polynomial nonlinearities that are perturbed by small, multiplicative, space-time white noise. The equations are defined on a one-dimensional bounded interval with Dirichlet…
We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover…
The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden attractor} in the case of multistability as well as a classical {self-excited…
In this paper, we will make use of the Gromov-Hausdorff distance between compact metric spaces to establish the continuous dependence and the Gromov-Hausdorff stability of global attractors for damped wave equations under perturbations of…
This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $\beta$-plane. Previous results in this setting by M.A.H.…
We discuss the estimates for the $L^p$-norms of systems of functions that are orthonormal in $L^2$ and $H^1$, respectively, and their essential role in deriving good or even optimal bounds for the dimension of global attractors for the…
This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing…
We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to…
We study the Chirikov (standard) map at large coupling $\lambda \gg 1$, and prove that the Lyapounov exponent of the associated Schroedinger operator is of order $\log \lambda$ except for a set of energies of measure $\exp(-c…
We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad…
We prove existence of global attractors for damped hyperbolic equations of the form $$\aligned \eps u_{tt}+\alpha(x) u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u),\quad x\in \Omega, t\in[0,\infty[, u(x,t)&=0,\quad x\in \partial…
In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$ \ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0, $$ under assumptions ensuring local convexity of the potential near equilibrium and…
We obtain Strichartz-type estimates for the fractional Schr\"odinger operator $f \mapsto e^{it(-\Delta)^{\gamma/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the…