Finite-dimensional behaviour of a nonautonomous partial differential equation: Forced oscillations of an extensible beam
We prove the existence of a finite-dimensional attractor for the nonautonomous equation: \[ U_{tt}+ dU_ t+ aU_{xxxx}+ \left( K(t)- \int_ 0^ \pi U_ z^ 2(z,t)dz\right) U_{xx}=f(x,t), \tag{1} \] \(U(0,t)=U(\pi,t)=U_{xx}(0,t)=U_{xx}(\pi,t)=0\), where \(K\) and \(f\) are time periodic with periods \(\omega_ 1\), \(\omega_ 2\). The equation (1) represents a model of transversal motion of a hinged beam where the external damping is proportional to the velocity.
existence of a finite-dimensional attractor
scientific article; zbMATH DE number 166902