On the Loss of Regularity in a Degenerate Vibrating Beam Equation

We study the well-posedness in Sobolev spaces \(H^s(\textbf{R})\) of the Cauchy problem for \(D_t^2u=y(1+D_y^2)^2yu\) , where \((t,y)\in [0,\infty )\times \textbf{R}\) . Our results show that solutions u(t, y) undergo infinite losses of derivatives \(D_y\) on a subset of positive time measure \(S\subset [0,T]\) for every \(T>\pi .\) We also find explicitly a basis of eigenfunctions for the degenerate elliptic operator which do not belong to any Sobolev space with index \(s\ge 5/2.\)