| Popis: |
The work deals with a generalised Timoshenko type system whose coefficients are supposed to be smooth functions depending on the spatial variable only.The aim of this study focuses on establishing some sufficient condition for well-posedness of the problem and proving periodicity behavior of the existing solution when the system takes into account the influence of an elastic medium. The technique, we develop in order to derive the aforementioned results, consists of introducing some Sobolev spaces and studying the properties of the system operator, when it is embedded in them. Moreover, there is a natural comparison between the local density of the energy conservation law of the physical clamped Timoshenko beam system and the correspondent inner product of the basic pivot space, where we are interested in studying the main properties of the abstract system operator. In this way, it is proven that the system operator is maximal monotone onto the basic pivot space. Moreover, it turns out to be an anti-self-adjoint one. As a consequence, it is proven that the whole spectrum of the regarded operator consists only of completely imaginary isolated points with a unique accumulation point. |
