On the problem of semiinfinite beam oscillation with internal damping

Abstract

We study the Cauchy problem for the equation of the form $$ \ddot{u}(t) + (\aa A + B)\dot{u}(t) + (A+G)u(t) = 0,\tag* $$ where $A$, $B$, and $G$ are \o s in a Hilbert space $\Cal H$ with $A$ selfadjoint, $\sigma(A)=[0,\infty)$, $B\ge0$ bounded, and $G$ symmetric and $A$-subordinate in a certain sense. Spectral properties of the correspondent operator pencil $L(\lambda) := \lambda^2I + \lambda (\alpha A + B) + A + G$ are studied, and existence and uniqueness of generalized and classical solutions of the Cauchy problem are proved. Equations of the type (*) include, e.g., an abstract model for the problem of semiinfinite beam oscillations with internal damping.