Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space

We study the Cauchy problem for damped wave equations with a fractional damping ( − Δ ) θ u t in R n . We derive more sharp decay estimates of the total energy based on the energy method in the Fourier space combined with the Haraux–Komornik inequality. Especially, in the case when 0 ≤ θ ≤ 1 / 2 the rate of decay of the total energy becomes almost optimal. The method in this paper can be applied to other equations and in particular it seems to be quite effective in the case of frictional dissipation, i.e., when θ = 0 .