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

Abstract

Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: rho u(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to(2))(gamma)Delta u + u(t) = 0, rho v(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to 2)(gamma)Delta v + v(t) = 0, with homogeneous Dirichlet boundary condition and rho > 0 and gamma > 0. When either the coefficient rho 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 rho and the initial data, we derive the decay estimates of the solutions and their second order derivatives.