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: ρ u ″ + ( ∫ Ω | A 1 / 2 u ( x , t ) | 2 d x ) γ A u + A u ′ = 0 . For all t ⩾ 0 , we will give the optimal decay estimate C − 1 ( 1 + t ) − 1 / γ ⩽ ‖ A 1 / 2 u ( 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 ‖ A u ( t ) ‖ 2 for t ⩾ 0 .