Related papers: Embedding of global attractors and their dynamics

The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as…

We study the asymptotic behavior of solutions of one coupled PDE-ODE system arising in mathematical biology as a model for the development of a forest ecosystem. In the case where the ODE-component of the system is monotone, we establish…

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz…

For an arbitrary unbounded domain $\Omega\subset\R^3$ and for $\eps>0$, we consider the damped hyperbolic equations \leqno{(H_\eps)} \eps u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u),\quad x\in \Omega,…

We consider an autonomous system of partial differential equations for one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases…

We prove that the weakly damped nonlinear Schr\"odinger flow in $L^2(\mathbb{R})$ provides a dynamical system which possesses a global attractor. The proof relies on the continuity of the Schr\"odinger flow for the weak topology in…

The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional…

In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed…

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical…

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are…

Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient…

This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness,…

The existence of a singularity by definition implies a preferred scale--the affine parameter distance from/to the singularity of a causal geodesic that is used to define it. However, this variable scale is also captured by the expansion…

We investigate fractional Cauchy type problem. By using Schauder fixed point theorem we obtain sufficient conditions for the global attractivity of solutions for nonlinear fractional differential equations in weighted spaces.

This paper is devoted to the quantitative study of the attractive velocity of generalized attractors for infinite-dimensional dynamical systems. We introduce the notion of~$\varphi$-attractor whose attractive speed is characterized by a…

In this work we consider the non local evolution equation with time-dependent terms which arises in models of phase separation in $\mathbb{R}^N$ \[ \partial_t u=- u + g \left(\beta(J*u) +\beta h(t,u)\right) \] under some restrictions on…

We study a two-phase modified Stefan problem modeling solid combustion and nonequilibrium phase transition. The problem is known to exhibit a variety of non-trivial dynamical scenarios. We develop a priori estimates and establish…

For stochastic Hindmarsh-Rose equations with additive noises in the study of neurodynamics, the longtime and global pullback dynamics on a two-dimensional bounded domain is explored in this work. Using the additive transformation and by the…

We consider a mathematical model coupling the Cahn-Hilliard system for phase separation with an additional equation describing the diffusion process of a chemical quantity whose concentration influences the physical process. The main…

We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This…