Decay estimates for a Klein-Gordon model with time-periodic coefficients (Q2662144)

In this paper, the author studies the energy decay estimates to the solutions of the Cauchy problem for the Klein-Gordon equation with positive time-periodic dissipation \(b(t)\) and mass \(m(t)\). There exist many papers in which decay estimates for the solution to wave models are investigated. Especially, in [Hiroshima Math. J. 38, No. 3, 397--410 (2008; Zbl 1172.35442)], the second author of this paper considered the linear Cauchy problem for a damped wave equation with time-periodic dissipation term \(b(t)\), and showed that the solution satisfies the well-known Matsumura-type estimate. The aim of this paper is to investigate how the presence of the mass term influences energy estimates with respect to the case of vanishing mass as above. The approach is based on a diagonalisation argument for high frequencies and a contradiction argument for bounded frequencies. For the entire collection see [Zbl 1457.35004].