Remarks on the existence and decay of the nonlinear beam equation (Q1326634)

We consider the following initial value problem for an abstract nonlinear beam equation \[ Ku_{tt}+ A^ 2 u+ M(\| A^{1/2} u \|^ 2) Au+ u_ t= 0, \qquad u(0)= u_ 0, \quad Ku_ t(0)= Ku_ 1, \] where \(A\) is a selfadjoint positive definite operator in a Hilbert space \({\mathcal H}\) with domain \(D(A)\) dense in \({\mathcal H}\) and the embedding of \(D(A^ r)\) into \(D(A^ s)\) is compact for \(r>s\geq 0\). We prove existence and decay of weak solutions.