Advanced Search

The operator $L_0:D_{L_0}\subset H\to H$, $L_{0}u=\frac{1}{r}\frac{d}{dr}\left\{r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)\right]\right\}$, $D_{L_0}=\{u\in C^4\left([0,R]\right),u^{\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=0,u\left(R\right)=u^{\text{'}}\left(R\right)=0\}$, $H=L_{2,r}(0,R)$ is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on $L_0$ (in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types $L_{0}u=g$ and $u_{tt}+L_{0}u=g$, respectively.

It is proved that any weak solution to a nonlinear beam equation is eventually globally oscillatory, i.e., there is a uniform oscillatory interval for large times.