Exponential decay of solutions of damped wave equations in one dimensional space in the $L^p$ framework for various boundary conditions

TL;DR

This paper proves that solutions to damped wave equations in one dimension decay exponentially under various boundary conditions, using hyperbolic system analysis and energy estimates.

Contribution

It introduces a unified approach to establish exponential decay for damped wave equations with space- and time-dependent damping coefficients across multiple boundary conditions.

Findings

Solutions decay exponentially in the L^p framework

Decay estimates hold for Dirichlet, Neumann, and dynamic boundary conditions

The method applies to damping coefficients varying in space and time

Abstract

We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.