Sharp Energy Decay and Indirect Stabilization for a Nonlinearly Coupled Wave–Diffusion System with Mixed Boundary Conditions

We study a wave–diffusion system on a bounded Lipschitz domain whose components satisfy the same mixed Dirichlet–Neumann boundary decomposition. The wave is semi-linear, while dissipation acts only through the diffusion component; stabilization of the wave is therefore indirect. For a coercive mixed Laplacian and a defocusing subcritical source, we prove global finite-energy well-posedness, an exact energy identity, uniqueness, D(A). The spectral asymptotics prove optimality of the exponent for general strong data. The result identifies the precise high-frequency obstruction to exponential decay and separates geometric coercivity, nonlinear well-posedness, and the diffusion-mediated stabilization mechanism. A one-dimensional mixed-boundary computation is included only as a reproducibility check of the spectral asymptotics. 2020 Mathematics Subject Classification. 35L70, 35K55, 35B40, 47D06, 93D20.