Different Boundary Problems Governed by the Dynamic and Stationary Operator Nonlinear Vibration of the Plates

Abstract

Bouzeghaya Fouzia, Merouani Boubakeur In this paper, we propose to study some nonlinear boundary problems for the dynamically modified operator by adding a viscosity term -αΔu'' to the nonlinear vibrations of the plates. The field of application for vibrating plates is extensive. To meet user needs, we have considered the geometric shape, the density of the material constituting the plate, the plate thickness, and Poisson's ratio. Once the problems have been posed, our approach then consists of transforming them into nonlinear problems of the hyperbolic type. In this work, we study six boundary value problems and we prove for each problem an existence and uniqueness theorem. Finally, we demonstrate the existence of a solution to the stationary problem using a variant of Brouwer's fixed point theorem.