Asymptotic parabolicity for strongly damped wave equations

Abstract: 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 $u(0)$, $u'(0)$.