Decay Rates and Domain Dependence of a Coupled Wave-Heat System with Spatially Dependant Heat Coefficients

Abstract: We study of the long-term behavior of a coupled wave-heat system. The system consists of a wave equation and a heat equation on two adjacent Lipschitz domains coupled by a common interface, with the heat equation being allowed to incorporate spatially dependant coefficients. We first establish strong asymptotic stability independent of the domains and coefficients. To this end, we employ the framework of closure relations, which reduces the spectral analysis of the coupled system to that of a wave equation. Secondly, we analyze non-uniform decay rates for classical solutions to the coupled system. Using methods from the theory of $C_0$-semigroups and a non-orthogonal decomposition of the state space, we reduce the problem to a residual estimate which is independent of the heat domain and heat coefficients, instead depending monotonically on the wave domain and the interface. With this, we are able to extend the known non-uniform decay rates to spatially dependent heat coefficients, yielding logarithmic decay under no assumptions, as well as polynomial decay in one dimensions or under the Geometric Control Condition.