... in infinite dimensions,” in Nonlinear DifferentialEquations, J. Hale and P. Martinez-Amores, Eds., vol. 132, pp. 1–42, Pitman, Boston, Mass, USA, 1985.Advances in Difference Equations 11Setting ... stands for L2-inner product on L2. In particular, V−1 H−1, V0 L2,andV1 H10. The injection Vs1→ Vs2is compactwhenever s1>s2. For further convenience, for s ∈ R, introduce ... this damping depends on t he type of couplingand the wave speeds. When the wave speeds are the same E G, the indirect damping isactually strong enough to induce exponential stability for the...