Global Existence and Asymptotic Behavior of Solutions to Partially Dissipative Hyperbolic Equations

In this paper, we investigate the initial value problem for the partially dissipative hyperbolic equations in three space dimensions. Under the small initial data condition, the global existence and asymptotic behavior of solutions are established. The proof is based on the coupled Green function estimates for weakly dissipative hyperbolic systems, the space-time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah and the continuous argument.