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Information × Registration Number 0212U002060, 0109U001459 , R & D reports Title Qualitative theory of dissipative infinitedimension systems of mathematical physics popup.stage_title Head Chyeshov Igor, Registration Date 18-01-2012 Organization Kharkov National University named after V.N. Karazin popup.description2 The object of the research are initial-boundary value problems for a system of coupled Berger equations, coupled system "nonlinear plate + fluid", viscothermoelastic nonlinear system with memory in both variables. The goal of the research is to investigate synchronization phenomenon for the problem of nonlinear oscillations of the coupled Berger plates using the notion of global attractor; to study asymptotical behaviour of solutions to the system of viscothermoelasticity and dependence of the attraction velocity of the system trajectories to an attractor on equilibria type; to find what happens with the solutions to the problem when memory kernels goes to delta-functions; to prove well-posedness of systems, which describe coupled oscillations of plate and fluid; to study asymptotical behaviour of these systems. Stage we prove existence and uniqueness of solution to the initial-boundary value problem for the system of the coupled Berger equations and establish, that the problem generates dissipative dynamical system which posesses compact global attarctor. Syncronization results are established for 1) the system of linearly damped equations with global coupling; 2) the system of two nonlinearly damped equations with symmetric global coupling; 3) the system ot two linearly damped equations with symmetric localized coupling. For each case 1)-3) it is established, that the attractor tends to an attractor of the projection of the coupled system on the kernel of coupling operator, when a parameter of coupling intensity tends to infinity. We also study structure of the attractors for certain cases of coupling matrix and find conditions, when syncronization takes place for large, but finite values of the parameter of coupling intensity. Stage we prove well-posedness of the system describing viscothermoelastic plate with memory in both variables, in which the heat conduction process is described by Gurtin-Pipkin or Corleman-Gurtin equation. It is established that under standard conditions on memory kernels and nonlinearity the system posesses compact global attractor of fininte fractal dimension, which consists of the set of equilibria of the system and the trajectories connecting them. We find the conditions under which the trajectories tend to the attractor exponentially fast. It is also proved, that the solutions to the system with memory tend to solutions of a correspondin system without memory (already with the classical equation of heat conduction), when memory kernels tend to delta-functions. Stage we consider two models, which describe interaction of fluid in a container with an elastic part of the containeer wall. Nevertheless, we manage to prove existence of global attractor for corresponding dinamical systems. This means that mechanism of energy transmission form the plate to the fluid is such that dissipation in the fluid is sufficient for stabilisation of the entire system. The structure of the attractor is studied. The methods of the research are theoretical considerations. The research is of theoretical nature. Its results can be applied for prediciton of long time behaviour of various vibration machines. SYSTEM OF COUPLED BERGER EQUATIONS, SYNCRONISATION AT THE LEVEL OF THE ATTRACTOR, UPPER SEMICONTINUITY OF ATTRACTOR FAMILY WITH RESPECT TO PARAMETER, VISCOTHERMOELASTICITY, EQUATIONS WITH MEMORY, THERMOELASTIC BERGER PLATE, SINGULAR LIMIT OF EQUATIONS, LINEARIZED NAVIER-STOCKS SYSTEM, VON KARMAN PLATE, COUPLED SYSTEM "FLUID-PLATE", GLOBAL ATTRACTOR, FRACTAL EXPONENTIAL ATTRACTOR.. Product Description popup.authors Набока О. Потьомкін М. Рижкова І. Щербина А. popup.nrat_date 2020-04-02 Close