Exponential decay for solutions to semilinear damped wave equation

TL;DR

This paper establishes exponential decay rates for solutions to a semilinear damped wave equation with linear damping in bounded domains, improving previous decay estimates by employing Lyapunov functions.

Contribution

It introduces a Lyapunov function approach to prove exponential decay for solutions with specific initial data, enhancing earlier results in the field.

Findings

Exponential decay of solutions under linear damping

Construction of Lyapunov functions for decay estimates

Improvement over previous decay rate results

Abstract

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in an article of Gazzola and Squassina.