Desigualdade Variacional da Equação Não-Linear Degenerada de Vibrações da Viga

ABSTRACT

In this article, we study the existence and uniqueness of a global weak solution for the degenerate nonlinear variational inequality

$||\begin{array}{l}K\left(x,t\right)u"+{\Delta }^{2}u+M\left({||u||}^2\right)\left(-\Delta u\right)+u\text{'}\ge f\\ u\left(0\right)=u_0\\ u\text{'}\left(0\right)=u_1\\ u_{{\lfloor }_{\Sigma }}=\frac{\partial u}{\partial \eta }{\lfloor }_{\Sigma }=0\end{array}$

where K(x, t) is a function defined on $Q=\Omega \times ]0,T[,K(x,t)\ge 0$ para todo $(x,t)\in Q$, M a continuous real function with specific properties and f belongs to the class of functions $L^2\left(0,T;H_0^1\left(\Omega \right)\right)$. We will use the Faedo-Galerkin method, monotone operator and Compactness to prove the existence and uniqueness of weak solutions.

Keywords:
penalty operator; variational inequality; Galerkin method