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

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.