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I will discuss well-posedness and long-time behavior of Mindlin-Timoshenko plate equations that describe vibrations of thin plates. This system of partial differential equations was derived by R. Mindlin in 1951 (though E. Reissner also considered an analogous model earlier in 1945). It can be regarded as a generalization of the Timoshenko beam model (1937) to flat plates, and is more accurate than the classical Kirchhoff-Love plate theory (1888) because it accounts for shear deformations.
I will present a semilinear version of the Mindlin-Timoshenko system. The primary feature of this model is the interplay between nonlinear frictional forces (``damping”) and nonlinear source terms. The sources may represent restoring forces, such as (nonlinear refinement on) Hooke's law, but may also have a destabilizing effect amplifying the total energy of the system, which is the primary scenario of interest.
The dissertation verifies local-in-time existence of solutions to this PDE system, as well as their continuous dependence on the initial data in appropriate function spaces. The global-in-time existence follows when the dissipative frictional effects dominate the sources. In addition, a potential well theory is developed for this problem. It allows us to identify sets of initial conditions for which global existence follows without balancing of the damping and sources, and sets of initial conditions for which solutions can be proven to develop singularities in finite time.
Advisers: Mohammad A. Rammaha and Daniel Toundykov