## Journals

- Advances in Mathematics of Communications
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- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

DCDS

This is a systematic study of order-preserving (or monotone) random
dynamical systems which are generated by
cooperative random or stochastic differential equations.
Our main results concern the long-term behavior of these systems, in
particular the existence of equilibria and attractors and
a limit set trichotomy theorem.
Several applications (models of the control of the protein synthesis
in a cell, of gonorrhea
infection and of symbiotic interaction in a random environment)
are treated in detail.

CPAA

We study long-time dynamics of a class of
abstract second order in time evolution equations
in a Hilbert space with the damping term depending both on displacement
and velocity. This damping represents the nonlinear strong dissipation
phenomenon perturbed with relatively compact terms.
Our main result states
the existence of a compact finite dimensional attractor.
We study properties of this attractor.
We also establish the existence of a fractal exponential
attractor and give the conditions that guarantee the existence
of a finite number of determining functionals.
In the case when the set of equilibria is finite and hyperbolic
we show that every trajectory is attracted by some equilibrium
with exponential rate.
Our arguments involve a recently
developed method based on the "compensated" compactness and
quasi-stability estimates. As an application we consider the
nonlinear Kirchhoff, Karman and Berger plate models with different
types of boundary conditions and strong damping terms. Our results
can be also applied to the nonlinear wave equations.

DCDS

We prove the existence of a compact, finite dimensional, global attractor
for a coupled PDE system comprising a nonlinearly damped semilinear wave
equation and a nonlinear system of thermoelastic plate equations,
without any mechanical (viscous or structural) dissipation in the
plate component.
The plate dynamics is modelled following Berger's approach; we investigate
both cases when rotational inertia is included into the model and when it
is not.
A major part in the proof is played by an estimate--known as
stabilizability estimate--which shows that the difference of any
two trajectories can be exponentially stabilized to zero, modulo a
compact perturbation.
In particular, this inequality yields bounds for the attractor's fractal
dimension which are independent of two key parameters, namely $\gamma$
and $\kappa$, the former related to the presence of rotational inertia
in the plate model and the latter to the coupling terms.
Finally, we show the upper semi-continuity of the attractor with
respect to these parameters.

DCDS-B

We consider a stochastically perturbed coupled system consisting of linearized 3D
Navier--Stokes equations in a bounded domain and the classical (nonlinear)
elastic plate equation for in-plane motions
on a flexible flat part of the boundary. This kind of models arises in
the study of blood flows in large arteries.
Our main result states the existence of a random pullback attractor
of finite fractal dimension. Our argument is based on some modification of the method of
quasi-stability estimates recently developed for deterministic systems.

EECT

We study dynamics of a coupled system consisting of the 3D
Navier--Stokes equations which is linearized near a certain
Poiseuille type flow between two unbounded circular cylinders and
nonlinear elasticity equations for the transversal displacements
of the bounding cylindrical shells.
We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.

CPAA

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 in time. This allows
us to show that the model considered
generates an evolution operator semigroup $S_t$ on a certain space of Lipschitz type
functions over delay time interval.
The operators $S_t$ are closed for all $t\ge 0$ and continuous for
$t$ large enough. Our main result shows that the semigroup $S_t$
possesses compact global and exponential attractors of finite
fractal dimension.
Our argument is based on the recently developed method of quasi-stability
estimates and involves some extension of the theory of global attractors for
the case of closed evolutions.

DCDS-B

We study invariance and monotonicity properties of Kunita-type
sto-chastic differential equations in $\mathbb{R}^d$ with delay.
Our first result provides sufficient conditions for the invariance
of closed subsets of $\mathbb{R}^d$.
Then we present a comparison principle and show that under appropriate conditions
the stochastic delay system considered generates a monotone
(order-preserving) random dynamical system. Several applications
are considered.

DCDS

We study dynamics of a class of nonlinear Kirchhoff-Boussinesq plate
models.
The main results of the paper are: (i)
existence and uniqueness of weak (finite energy) solutions, (ii)
existence of weakly compact attractors.

DCDS

We consider non-linear parabolic stochastic partial differential
equations with dynamical boundary conditions and with a noise
which acts in the domain but also on the boundary and is
presented by the temporal generalized derivative of an infinite
dimensional Wiener process.
We prove that solutions to this stochastic partial differential equation
generate a random dynamical system. Under additional conditions we
show that this system is monotone.
Our main result states the existence of a compact global
(pullback) attractor.

EECT

We deal with an initial boundary value problem for the
Schrödinger-Boussinesq system
arising in plasma physics in two-dimensional domains.
We prove the global Hadamard well-posedness of this problem
(with respect to the topology which is weaker than topology associated
with the standard variational (weak) solutions)
and study properties of
the solutions. In the dissipative case the existence of a global attractor
is established.

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