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### Open Access Journals

CPAA

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space
in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of
arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation
where, in addition, the dissipativity of the associated solution semigroup is established.

DCDS-B

We give a comprehensive study of strong uniform attractors of nonautonomous
dissipative systems for the case where the external forces are not
translation compact. We introduce several new classes of external forces that
are not translation compact, but nevertheless allow the attraction in a strong
topology of the phase space to be verified and discuss in a more detailed way
the class of so-called normal external forces introduced before. We also develop
a unified approach to verify the asymptotic compactness for such systems based
on the energy method and apply it to a number of equations of mathematical
physics including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains.

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.

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