Sharp polynomial decay rate of multi-link hyperbolic-parabolic systems with different network configurations

The asymptotic behavior of hyperbolic -parabolic coupled systems on some special networks (loop -type, chain -type, tree -type) are discussed. By a detailed spectral analysis, the asymptotic expressions of the spectrum of these three networks are obtained, and based on which, the lack of exponential decay is deduced for all these systems. Then using the frequency domain approach together with the asymptotic expressions of the spectrum, we derive the optimal polynomial decay rate of the systems under the smooth initial states. Moreover, some numerical simulations are given to support the results obtained.