Stabilization of the wave equation with external force

TL;DR

This paper investigates how the energy of solutions to the wave equation diminishes over time when subjected to localized damping and external forces, linking decay rates to a specific forced differential equation.

Contribution

It introduces a method to determine energy decay rates for wave equations with external forces using a related forced differential equation.

Findings

Decay rates depend on the solution of a specific forced differential equation.

Energy diminishes at rates characterized by the derived differential equation.

The approach provides a new way to analyze stabilization in wave equations with external influences.

Abstract

We study the rate of decay of the energy functional of solutions of the wave equation with localized damping and a external force. We prove that the decay rates of the energy functional is determined from a forced differential equation.