Related papers: Dissipative flows, global attractors and shape the…
There exists a variety of physically interesting situations described by continuous maps that are nondifferentiable on some surface in phase space. Such systems exhibit novel types of bifurcations in which multiple coexisting attractors can…
The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces…
In this work we introduce the concept of generalized exponential $\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$ is a universe of families in $X$, which is a compact and positively invariant family that…
This paper concerns the dynamical behavior of weakly reversible, deterministically modeled population processes near the facets (codimension-one faces) of their invariant manifolds and proves that the facets of such systems are "repelling."…
We study dynamics of a generic quadratic diffeomorphism, a 3D generalization of the planar H\'{e}non map. Focusing on the dissipative, orientation preserving case, we give a comprehensive parameter study of codimension-one and two…
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…
We consider dynamical behavior of non-autonomous wave-type evolutionary equations with nonlinear damping, critical nonlinearity, and time-dependent external forcing which is translation bounded but not translation compact (i.e., external…
This paper is concerned with the Morse theory of attractors for semiflows on complete metric spaces. First, we construct global Morse-Lyapunov functions for Morse decompositions of attractors. Then we extend some well known deformation…
Partial Differential Equations (PDEs) play a crucial role as tools for modeling and comprehending intricate natural processes, notably within the domain of biology. This research explores the domain of microbial activity within the complex…
We consider the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of…
We study the question of global attraction, in the energy norm, for finite energy solutions to classical particle coupled to a scalar wave field. The attraction could take place to either the set of the stationary solutions in the case of a…
This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.
We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic PDE \begin{equation*}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) %\tag{$*$} \end{equation*} on the unit…
We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the…
In this article we initiate the mathematical study of the dynamics of a system of nonlinear Partial Differential Equations modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic…
We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is…
We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom…
The Conley index theory is a powerful topological tool for obtaining information about invariant sets in continuous dynamical systems. A key feature of Conley theory is that the index is robust under perturbation; given a continuous family…
We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow in an (unbounded) domain and a classical (possibly nonlinear) elastic plate equation for transversal…
In this paper, we present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear…