Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients

TL;DR

This paper investigates conditions under which solutions to a dissipative hyperbolic equation with time-dependent coefficients asymptotically resemble solutions of the free wave equation, focusing on energy non-decay scenarios.

Contribution

It establishes a necessary and sufficient condition on the coefficient for solutions to approach free wave solutions in a non-decaying energy context.

Findings

Identifies a specific condition on the coefficient for asymptotic equivalence

Demonstrates the property under energy non-decay assumptions

Provides theoretical criteria for solution behavior in hyperbolic equations

Abstract

This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.