Global Existence and Exponential Stability of a Boundary Contact Problem for the 2D Mindlin–Timoshenko System

Abstract

We analyze a contact problem for the two-dimensional Mindlin–Timoshenko system describing the vibratory motion of elastic plates in unilateral contact with a rigid obstacle along a portion of the boundary. Precisely, we establish the existence of solutions and prove that the energy of the system decays exponentially to zero as time approaches infinity. The existence proof relies on a penalization method: we construct solutions to the original contact problem as the limit of a sequence of solutions to a family of penalized systems. The exponential energy decay of the contact problem (as t→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow \infty $$\end{document}) is then obtained as the uniform limit of the corresponding exponential decay estimates for the penalized systems.