A lower decay estimate for a degenerate Kirchhoff type wave equation with strong dissipation

Consider the initial-boundary value problem for the degenerate Kirchhoff type wave equation with strong dissipation : ρ ∂2u/∂t2 − (∫ Ω |∇u(x,t)|2 dx ) Δu−δΔ ∂u /∂t = 0. For all t ≥ 0, a lower decay estimate of the solution ∥∇u(t)∥2 ≥ c (1 + t)−1 is derived when either the coefficient ρ or the initial data are appropriately smaller than the coefficient δ