Global Existence and Decay Properties of Solutions for Coupled Degenerate Dissipative Hyperbolic Systems of Kirchhoff Type

Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: ρ<i>u</i>_<i>tt</i> − (||∇<i>u</i>(<i>t</i>)||² + ||∇<i>v</i>(<i>t</i>)||²)^γΔ<i>u</i> + <i>u</i>_<i>t</i> = 0, ρ<i>v</i>_<i>tt</i> − (||∇<i>u</i>(<i>t</i>)||² + ||∇<i>v</i>(<i>t</i>)||²)^γΔ<i>v</i> + <i>v</i>_<i>t</i> = 0, with homogeneous Dirichlet boundary condition and ρ > 0 and γ > 0. When either the coefficient ρ or the initial data are appropriately small, we prove the global existence theorem by using several identities and the energy decay. Moreover, under the same assumption for ρ and the initial data, we derive the decay estimates of the solutions and their second order derivatives.