#### Vol. 11, No. 2, 2018

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Asymptotic limits and stabilization for the 2D nonlinear Mindlin–Timoshenko system

### Fágner Dias Araruna, Pablo Braz e Silva and Pammella Queiroz-Souza

Vol. 11 (2018), No. 2, 351–382
##### Abstract

We show how the so-called von Kármán model can be obtained as a singular limit of a Mindlin–Timoshenko system when the modulus of elasticity in shear $k$ tends to infinity. This result gives a positive answer to a conjecture by Lagnese and Lions in 1988. Introducing damping mechanisms, we also show that the energy of solutions for this modified Mindlin–Timoshenko system decays exponentially, uniformly with respect to the parameter $k$. As $k\to \infty$, we obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.

 Dedicated to Enrique Fernández-Cara on the occasion of his 60th birthday

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##### Keywords
vibrating plates, Mindlin–Timoshenko system, von Kármán system, singular limit, uniform stabilization
##### Mathematical Subject Classification 2010
Primary: 35Q74, 74K20, 35B40
##### Milestones
Received: 21 July 2016
Revised: 5 May 2017
Accepted: 5 September 2017
Published: 17 October 2017
##### Authors
 Fágner Dias Araruna Departamento de Matemática Universidade Federal da Paraíba João Pessoa, PB Brazil Pablo Braz e Silva Departamento de Matemática Universidade Federal de Pernambuco Recife, PE Brazil Pammella Queiroz-Souza Unidade Acadêmica de Matemática Universidade Federal de Campina Grande Campina Grande, PB Brazil