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

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.