Scattering and modified scattering for abstract wave equations with time-dependent dissipation

TL;DR

This paper investigates abstract wave equations with weak, time-dependent dissipation, demonstrating how solutions relate to free solutions modulated by decay functions, extending prior work on dissipation effects.

Contribution

It provides a new analysis of how solutions to dissipative wave equations relate to free solutions under specific conditions on dissipation, expanding understanding of dissipation effects.

Findings

Solutions are closely related to free solutions multiplied by decay functions.

Conditions on dissipation coefficients ensure this relation holds.

Extends previous work by connecting dissipation to solution behavior.

Abstract

We consider the initial-value problem of abstract wave equations with weak dissipation. We show that under conditions on the dissipation coefficient and its derivative the solutions to the abstract dissipative equation are closely related to solutions of the free problem multiplied by a decay function. This paper gives the counterpart to a recent paper of T.Yamazaki [Adv. Differential Equ., 11(4):419--456, 2006], where effective dissipation terms and the relation to the corresponding abstract parabolic problem are considered.