Trajectory Behavior of Weak Solutions of the Piezoelectric Problem with Discontinuous Interaction Function on the Phase Variable

Павло Олегович Касьянов, Лілія Сергіївна Палійчук


The autonomous second order inclusion in a bounded domain, which is modeling the behavior of a class of the controlled piezoelectric fields with nonmonotonous potential, is studied. The investigated system describes not only controlled piezoelectric process with multivalued law “reaction-displacement”, but a wide class of controlled processes of Continuum Mechanics. Conditions on the parameters of the problem do not guarantee the uniqueness of solution of the corresponding Cauchy problem. In particular, any conditions on continuity, monotony of the nonlinear term by a phase variable are not assumed. We study the dynamics of weak solutions of the investigated problems in terms of the theory of trajectory and global attractors for multivalued semiflows generated by weak solutions of given problem. By using the well-known abstract results on the existence of trajectory attractor in the space of trajectories, we show the existence of trajectory attractor in the extended phase space for solutions of the considered evolution problem. Its structural properties are studied. Its relationship with the global attractor and space of complete trajectories is provided. Obtained results are applied to the mathematical model which describes the dynamics of the piezoelectric process.


Trajectory attractor; Controlled piezoelectric field


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