Asymptotic parabolicity for strongly damped wave equations

TL;DR

This paper investigates the long-term behavior of solutions to strongly damped wave equations involving a selfadjoint operator, showing that solutions asymptotically split into a dominant part satisfying a first-order equation and a negligible remainder.

Contribution

It establishes the asymptotic parabolicity of solutions to a class of strongly damped wave equations with a general positive selfadjoint operator and provides a canonical way to determine initial conditions for the dominant component.

Findings

Solutions decompose into a main part and a negligible remainder as time progresses.

The main part satisfies a first-order damping equation.

Initial conditions for the dominant component are explicitly characterized.

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