We study the equations describing the motion of a thermal non-equilibrium gas with one non-equilibriummode. In three space dimensions it is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that Shizuta-Kawashima criterion is violated. However, there is a significant difference between one dimensional and three dimensional flows in how the criterion is violated. As a consequence, the velocity components in their solutionsbehave differentlywhile thermal dynamic variables share common properties.