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DCDS

The equations describing planar motion of a homogeneous,
incompressible generalized Newtonian fluid are considered.
The stress tensor is given constitutively as
$\T=\nu(1+\mu|\Du|^2)^{\frac{p-2}2}\Du$, where $\Du$ is the
symmetric part of the velocity gradient.
The equations are complemented by periodic boundary conditions.

For the solution semigroup the Lyapunov exponents are computed using a slightly generalized form of the Lieb-Thirring inequality and consequently the fractal dimension of the global attractor is estimated for all $p\in(4/3,2]$.

For the solution semigroup the Lyapunov exponents are computed using a slightly generalized form of the Lieb-Thirring inequality and consequently the fractal dimension of the global attractor is estimated for all $p\in(4/3,2]$.

DCDS-B

A nonlinear reaction-diffusion problem with a general, both
spatially and delay distributed reaction term is considered in an unbounded
domain $\mathbb{R}^N$. The existence of a unique weak solution is proved. A locally
compact attractor together with entropy bound is also established.

CPAA

The equations of an incompressible, homogeneous fluid
occupying a bounded domain in $\mathbb R^3$ are considered.

The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.

The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.

CPAA

We give an explicit estimate of the fractal dimension
of the global attractor to the wave
equation with nolinear damping.
The nonlinearities are smooth functions
of certain polynomial growth.
As a by-product we estimate the dimension of
the exponential attractor for the time $\tau$
solution operator provided that $\tau$ is
sufficiently large.
The main tool used in the proof
is the so-called method of the trajectories.

keywords:
nonlinear damping.
,
global attractor
,
fractal dimension
,
wave equation
,
exponential attractor

DCDS-S

We study the impact of an oscillating external force on the motion
of a viscous, compressible, and heat conducting fluid. Assuming
that the frequency of oscillations increases sufficiently fast
as the time goes to infinity, the solutions are shown to stabilize
to a spatially homogeneous static state.

PROC

We study the generalized logistic equation where the feedback is captured
by the time convolution with a nonnegative measure and the diffusion is the laplacian
plus the p-laplacian with $p >= 2$. We prove that the equation has an exponential
attractor provided that the solutions are asymptotically bounded.

CPAA

We consider two-dimensional flows of an incompressible non Newtonian
fluid where the departure from the Navier-Stokes fluid is due to the
viscosity depending on both the rate of deformation and the
pressure. We assume that the resulting extra-stress is uniformly
elliptic and its derivative with respect to pressure is bounded in a
proper manner. Considering just the spatially-periodic setting, one
can prove the global existence and uniqueness of the strong
solution. Using the so-called method of trajectories, we also prove
the existence of an exponential attractor and estimate its fractal
dimension in terms of the data of the equation.

CPAA

We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's $\varepsilon$-entropy is also established using the method of trajectories.

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