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

抄録

Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: ρu_tt − (||∇u(t)||² + ||∇v(t)||²)^γΔu + u_t = 0, ρv_tt − (||∇u(t)||² + ||∇v(t)||²)^γΔv + v_t = 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.