ASYMPTOTIC BEHAVIOR FOR A GENERAL CLASS OF HOMOGENEOUS SECOND ORDER EVOLUTION EQUATIONS IN A HILBERT SPACE

We study the asymptotic behavior of solutions to the following general homogeneous second order evolution equation, with suitable assumptions on $p(t)$ and $r(t)$,

$$ \left\{\begin{array}{ll} p(t)u^{\prime\prime}(t) + r(t)u^{\prime}(t) \in Au(t) & \text{a.e. on}\ R^+,\\ u(0) = u_0, & \sup_{t\geq 0} |u(t)| < +\infty, \end{array} \right.$$

where $A$ is a maximal monotone operator in a real Hilbert space, and present some applications. In the homogeneous case, our results extend those given in [7, 10, 12].