Asymptotic parabolicity for strongly damped wave equations

For $S$ a positive selfadjoint operator on a Hilbert space, $$\frac{d^2u}{dt}(t) + 2 F(S)\frac{du}{dt}(t) + S^2u(t)=0$$ describes a class of wave equations with strong friction or damping if $F$ is a positive Borel function. Under suitable hypotheses, it is shown that $$u(t)=v(t)+ w(t)$$ where $v$ satisfies $$2F(S)\frac{dv}{dt}(t)+ S^2v(t)=0$$ and $$\frac{w(t)}{\|v(t)\|} \rightarrow 0, \; \text{as} \; t \rightarrow +\infty.$$ The required initial condition $v(0)$ is given in a canonical way in terms of $u(0)$, $u'(0)$.