Phragmén–Lindelöf Alternative Results for the Thermoelasticity of Type III on an Exterior Region in ℝ3

This paper investigates the spatial asymptotic behaviour of solutions to a coupled thermoelastic system of Green–Naghdi Type III in an exterior domain of R3. The system couples elastodynamics with a second-order heat conduction law and contains indefinite cross-coupling terms between mechanical and thermal variables. By constructing a weighted energy functional and deriving a first-order differential inequality in the radial direction, we establish a Phragmén–Lindelöf alternative: for each fixed time, the total energy either grows exponentially or decays exponentially as r→∞, with an explicit decay rate that depends on the material coefficients and a free parameter ω. This dichotomy itself reveals a fundamental symmetry in the spatial behaviour—growth versus decay—which is intimately linked to the radial symmetry of the exterior geometry and the inherent structure of the coupled system. The result provides a complete characterization of spatial stability for this thermoelastic model in unbounded exterior domains.