This paper is devoted to the study of the asymptotic behavior of the solutions of the system of equations that models the heat conduction with two temperatures. That is, we consider a mixture of isotropic and homogeneous rigid solids. We analyze the static problem in a semi-infinite cylinder where every material point has two temperatures with nonlinear boundary conditions on the lateral side. A Phragmén–Lindelöf alternative for the solutions is obtained by means of energy arguments. Estimates for the decay and growth of the solutions are presented. We also prove that the only solution vanishing in the exterior of a bounded set is the null solution for a particular subfamily of problems. Cone-like domains are considered in the last section, and we obtain decay estimates for the solutions when the total energy is bounded.

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