Lower Decay Estimates for Non-Degenerate Kirchhoff Type Dissipative Wave Equations

We consider the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations ρu′′+ a (∥A1/2u(t)∥2) Au + u′ = 0 and (u(0), u′(0)) = (u0, u1), where u0 ≠ 0. We derive the lower decay estimate ∥u(t)∥2 ≥ Ce−βt for t ≥ 0 with β > 0 for the solution u(t).