Asymptotic parabolicity for strongly damped wave equations

For $S$ a positive selfadjoint operator on a Hilbert space, $$\frac{d^{2}u}{dt}(t)+2F(S)\frac{du}{dt}(t)+S^{2}u(t)=0$$ 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 $$2F(S)\frac{dv}{dt}(t)+S^{2}v(t)=0$$ and $$\frac{w(t)}{\mathrm{\parallel }v(t)\mathrm{\parallel }}\to 0,\text{as}t\to +\mathrm{\infty }\mathrm{.}$$ The required initial condition $v(0)$ is given in a canonical way in terms of $u(0)$, $u^{\mathrm{\prime }}(0)$.