Spectral analysis and transport of energy density for an hyperbolic system with periodic oscillations

We deal with an hyperbolic degenerate system with periodic oscillations in the lower order term and principal part with characteristics of variable multiplicity. In the homogenization limit the effective system does not present a memory term, however it loses the property of energy conservation. A spectral analysis on the Bloch space is worked out for the system in some particular case of oscillating term. Through a decomposition in Bloch waves we determine the transport for the limiting energy density related to the solution a Cauchy problem with oscillatory initial data.