Polynomial stability of piezoelectric beams with magnetic effect and tip body. (English) Zbl 07885409
Summary: In this paper, we consider a dissipative system of one-dimensional piezoelectric beam with magnetic effect and a tip load at the free end of the beam, which is modeled as a special form of double boundary dissipation. Our main aim is to study the well-posedness and asymptotic behavior of this system. By introducing two functions defined on the right boundary, we first transform the original problem into a new abstract form, so as to show the well-posedness of the system by using Lumer-Phillips theorem. We then divide the original system into a conservative system and an auxiliary system, and show that the auxiliary problem generates a compact operator. With the help of Weyl’s theorem, we obtain that the system is not exponentially stable. Moreover, we prove the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455-478).
© 2024 Wiley-VCH GmbH.
© 2024 Wiley-VCH GmbH.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
| 47D06 | One-parameter semigroups and linear evolution equations |
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