ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Attractors of the Dynamical Systems Connected to the Parabolic Equation


Lebedev A. V

Faculty of mathematics and mechanics
Department of Differential Equations
St. Petersburg State University
Bibliotechnaya pl.2, Petrodvoretz, Saint Petersburg,198904,Russia


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.
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