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### Open Access Journals

DCDS

Our aim in this article is to construct exponential attractors for
singularly perturbed damped wave equations that are continuous with
respect to the perturbation parameter. The main difficulty comes from
the fact that the phase spaces for the perturbed and unperturbed
equations are not the same; indeed, the limit equation is a
(parabolic) reaction-diffusion equation. Therefore, previous
constructions obtained for parabolic systems cannot be applied
and have to be adapted. In particular, this necessitates a
study of the time boundary layer in order to estimate the difference
of solutions between the perturbed and unperturbed equations. We note
that the continuity is obtained without time shifts that have been used
in previous results.

CPAA

The paper is devoted to study of the longtime behavior
of solutions of a damped semilinear wave equation
in a bounded smooth domain of $\mathbb R^3$
with the nonautonomous
external forces and with the critical cubic growth rate of the
nonlinearity. In contrast to the previous papers, we prove the
dissipativity of this equation in higher energy spaces $E^\alpha$,
$0<\alpha\le 1$, without the usage of the dissipation integral
(which is infinite in our case).

DCDS

We prove that the attractor of the
1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive
space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations
in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof
for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed
to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones
for all time. The temporal evolution of the well-separated multi-soliton structures is described
by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons.
We develop a version of normal hyperbolicity theory for the weakly coupled LDS's with continuous time
and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in
discrete-time LDS's. While the LDS part of the theory may be of independent interest, the main difficulty
addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to
the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial
or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other
and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time
chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding
space-time entropy is strictly positive.

CPAA

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations
with a nonlinearity of arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal
regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the
nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with
the Navier-Stokes inertial term are also considered.

CPAA

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.

CPAA

In this paper we obtain sharp upper estimates on
the uniform Lyapunov dimension of a cascade system in terms of the
corresponding Lyapunov exponents of their components. The obtained
result is applied for estimating the Lyapunov and fractal dimensions
of the attractors of nonautonomous dissipative systems generated
by PDEs of mathematical physics.

CPAA

We apply the dynamical approach to the study of the second order
semi-linear elliptic boundary value problem
in a cylindrical domain
with a small parameter $\varepsilon$ at the second derivative with respect to
the variable $t$ corresponding to the axis of the cylinder.
We prove that, under natural assumptions on the nonlinear interaction
function $f$ and the external forces $g(t)$, this problem possesses
the uniform attractor $\mathcal A_\varepsilon$ and that these attractors tend
as $\varepsilon \to 0$ to the attractor $\mathcal A_0$ of the limit parabolic
equation. Moreover, in case where the limit attractor $\mathcal A_0$ is
regular, we give the detailed description of the structure of
the uniform attractor $\mathcal A_\varepsilon$, if $\varepsilon>0$ is small enough, and
estimate the symmetric distance between the attractors $\mathcal A_\varepsilon$
and $\mathcal A_0$.

CPAA

In this note, we establish a general result on the existence of global attractors
for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong
continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that
$S(t)$ be a closed map.

DCDS

Our aim in this paper is to study the Cahn-Hilliard equation with
singular potentials and dynamic boundary conditions. In particular, we
prove, owing to proper approximations of the singular potential and a
suitable notion of variational solutions, the existence and uniqueness
of solutions. We also discuss the separation of the solutions from the
singularities of the potential. Finally, we prove the existence of
global and exponential attractors.

CPAA

The paper deals with the Navier-Stokes equations in a strip in
the class of spatially non-decaying (innite-energy) solutions belonging to the
properly chosen uniformly local Sobolev spaces. The global well-posedness
and dissipativity of the Navier-Stokes equations in a strip in such spaces has
been rst established in [22]. However, the proof given there contains a rather
essential error and the aim of the present paper is to correct this error and to
show that the main results of [22] remain true.

## Year of publication

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