Decay rates for star-shaped degenerate heat-wave coupled networks

Abstract

This work investigates the long-time dynamics of a star-shaped network composed of degenerate heat and wave equations. The well-posedness of the system is proved by standard semigroup theories and a comprehensive criterion for strong stability in such degenerate partial differential equations (PDE) networks is established. Through frequency domain analysis, the polynomial decay rate is explored in two scenarios: networks with a single wave equation, where the explicit decay rate depends solely on the degree of degeneration in those diffusion coefficients of the heat parts, and networks with multiple wave equations, where the explicit decay rates are derived under specific irrationality conditions on the spatial lengths of the wave equations involved in the network using Diophantine approximation arguments. Finally, a generalized slow decay rate is derived, providing a broader understanding of the long-time behavior of this complex degenerate heat-wave networks.