Time Decay for Some Degenerate Hyperbolic Systems with Strong Dissipations

Consider the initial-boundary value problem for the coupled degenerate strongly damped hyperbolic system of Kirchhoff type with a homogeneous Dirichlet boundary condition. We give the polynomially decay estimates of the solutions and their derivatives. Moreover, when either the wave coefficient or the initial data are appropriately small, we derive a lower decay rate for the solutions.