Search academic texts

Information × Registration Number 0209U006045, 0106U001535 , R & D reports Title Asymptotic and qualitative behaviour of solutions to dissipative evolution equations with partial derivatives popup.stage_title Head Chueshov I., Registration Date 19-01-2009 Organization Kharkov National University named after V.N.Karazin popup.description2 Research report, p. 37, references 46. Objects of research are qualitative methods of research of asymptotical and qualitative behaviour of solutions to initial-boundary value problems of mathematical physics and potential theory methods. There are studied asymptotical behaviour of solutions to problems of thermoelasticity in various settings, namely, systems for thermoelastic Mindlin-Timoshenko plate and some coupled systems, that describe interaction of a thermoelastic plate and gas media. Well-posedness of number of linear problems of oscillations of piecewise-homogeneous plates with transversal shear effect is studied by potential theory methods. The goal of the work is to prove existence of compact global attractor for thermoelastic problems of Mindlin-Timoshenko, to prove stabilisation of solutions to a problem of oscillations of a thermoelastic von Karman plate in a subsonic gas flow, to prove existence of the attractor for a nonlinear system that describes acoustic camera and to study its properties, to reduce initial-value problems of oscillations of plates to systems of dynamical pseudodifferential boundary equations and to prove there well-posedness. Existence of global attractor for thermoelastic Mindlin-Timoshenko systems means that solutions to these problems uniformly tend to some bounded compact set. Finite fractal dimension of attractor means that asymptotic behaviour of solutions can be described by finite set of parameters. Also for an acoustic camera it was proved upper semicontinuity of family of attractors with respect to parameters that describe rotational inertial of plate filaments and intensity of the interaction of plate and gas, when one of the parameters (or both) tens to zero. This means that corresponding system without coupling (without rotational inertia) describes object behaviour quite accurate when intensity of interaction (rotational inertia) is small. As for the problem of oscillations of von Karman plate in the subsonic gas flow, the stabilization of entire systemis established. That is, it is shown that the triple of functions "plate displacement + plate temperature + perturbation velocity of the gas" - the solution to the system - tends to the set of stationary points of the problem, while time tends to infinity. Representation of solutions to problems of oscillations of piecewise-homogeneous thermoelastic plates with transversal shear effect as a sum of single and double layer potentials is found. This representation yields to a system of pseudodifferential boundary equations with respect to unknown densities of potentials; thus, dimension of the problem is decreased by one. Existence, uniqueness, and stability of solutions to such systems are proved in as scale of Sobolev-type spaces. Method of the research is theoretical reasoning. The work is of theoretical nature. Its results can be used for constructing effective algorithms of approximate study of asymptotical behaviour of nonlinear systems of thermoelasticity and aerothermoelasticity and oscillations of thermoelastic plates with transversal shear effect. DYNAMICAL SYSTEMS, GLOBAL ATTRAKTORS, REGULARITY, STABILIZATION, THERMOELASTICITY, AEROELASITCITY, BOUNDARY DIFFERENTIAL EQUATIONS, DYNAMICAL POTENTIALS, TRANSVERSAL SHEAR EFFECT. Product Description popup.authors popup.nrat_date 2020-04-02 Close