On decay properties of solutions for degenerate strongly damped wave equations of Kirchhoff type

We consider the initial-boundary value problem for the degenerate strongly damped wave equations of Kirchhoff type: . For all t?0, we will give the optimal decay estimate C−1(1+t)^−1/γ?‖A1/2u(t)^‖2?C(1+t)^−1/γ, when either the coefficient ρ is appropriately small or the initial data are appropriately small. And, we will show a decay property of the norm ‖Au(t)^‖2 for t?0.