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

We study on 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 and the nonlocal nonlinear term a(M) = 1+Mγ with γ > 0. Under the suitably smallness condition, we derive the upper decay estimates of the solution u(t) for the case of 0 < γ < 1 in addition to γ ≥ 1.