Large-time behavior of solutions to hyperbolic-elliptic coupled systems (Q1879427)

The authors continue their study of the hyperbolic-elliptic system derived in [Monogr. Surv. Pure Appl. Math. 99, 87--127 (1998; Zbl 0942.35048)]. They generalize their previous results and prove the global existence and the asymptotic decay of solutions for small initial data in a suitable Sobolev space. Moreover, they show that solutions are well-approximated, for large time, by solutions to the corresponding hyperbolic-parabolic coupled system. The results are obtained by deriving \textit{a priori} estimates as well as by the spectral analysis of solutions to the linearized problem.