## Attractors of the Dynamical Systems Connected to the Parabolic Equation

### Author(s):

**Lebedev A. V**

Faculty of mathematics and mechanics

Department of Differential Equations

St. Petersburg State University

Bibliotechnaya pl.2, Petrodvoretz, Saint Petersburg,198904,Russia

Andrey.Lebedev@pobox.spbu.ru

### Abstract:

In the PhD thesis for the first time there are investigated the
qualitative properties of the dynamical and semidynamical systems
generated by the discretization of the Dirichlet problem for the
parabolic equation by using Adams method of the arbitrary degree.

The sufficient conditions of the dynamical system construction are obtained.
The sufficient condition is given under which the system is dissipative
and the upper estimation of the diameter of the global attractor of the system is obtained.
The upper estimations of the Hausdorff dimension of the global attractor of the system
are obtained for both cases of small and large Lipschitz constants of the nonlinearity.
The obtained estimate of the Hausdorff dimension does not depend on the
parameters of the approximation method.
New results concerning the behaviour of the trajectories of gradient-like system
of the differential equations generated by the restriction of Chafee-Infante system
on its inertial manifold in the case of critical parameter value are obtained.

The global polynomial estimation of the rate of attraction of
the trajectories to the attractor in the terms of the starting approximation is proved.
The logarithmic approximation of the deviation of the attractor of the perturbed
system from the attractor of the source system in the terms of the value of system
perturbation is obtained.