Optimal energy decay rates for Klein-Gordon equations with Kelvin-Voigt damping

TL;DR

This paper investigates the long-term energy decay of solutions to a 1D Klein-Gordon equation with Kelvin-Voigt damping, establishing convergence to zero and optimal polynomial decay rates.

Contribution

It proves that all solutions' energy tends to zero and derives the optimal polynomial decay rate for solutions, addressing spectral challenges on the imaginary axis.

Findings

Energy of solutions converges to zero over time.

Established optimal polynomial decay rates.

Spectral analysis reveals multiple spectral points on the imaginary axis.

Abstract

We study the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. One of the interesting features of the equation is that the generator of the associated $C_0$-semigroup has multiple spectral points on the imaginary axis. As our main result, we show that the energy of every possible solution converges to zero as time goes to infinity and, moreover, we provide an optimal polynomial energy decay rate for a certain class of solutions.