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 , 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