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CPAA

We study asymptotic dynamics of a coupled system consisting of linearized 3D
Navier--Stokes equations in a bounded domain and a classical (nonlinear)
elastic plate equation for transversal displacement
on a flexible flat part of the boundary.
We show that this problem generates a semiflow on appropriate
phase space.
Our main result states the existence of a compact
finite-dimensional global attractor for this semiflow.
We do not assume any kind of mechanical damping in the plate component.
Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system.
To achieve the result we first study the corresponding linearized model
and show that this linear model generates strongly continuous
exponentially stable semigroup.

CPAA

We prove well-posedness of energy type solutions to an interacting
system consisting of the 3D linearized Euler equations and a
(possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary.
In the damped case under some conditions concerning the plate nonlinearity
we prove the existence of a compact global attractor for the corresponding
dynamical system and describe the situations when this attractor has a finite
fractal dimension.

DCDS

The existence and uniqueness of solutions for a stochastic
reaction-diffusion equation with infinite delay is proved.
Sufficient conditions ensuring stability of the zero solution are
provided and a possibility of stabilization by noise of the
deterministic counterpart of the model is studied.

DCDS-B

We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.

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