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) +
(åA + B)\dot{u}(t) + (A+G)u(t) = 0,\tag*
where
A
,
B
, and
G
are øs in a Hilbert space $\Cal H$ with
A
selfadjoint,
σ(A)=[0,∞)
,
B≥0
bounded, and
G
symmetric and
A
-subordinate in a certain sense. Spectral properties of the correspondent operator pencil
L(λ):=
λ
2
I+λ(αA+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.