In this paper we consider the linear theory of thermoelastic solids characterized by Cattaneo-Maxwell's constitutive relation for the heat flux in the form proposed by Pao and Banerjee (Pao, Y. H. and Banerjee, D. K., Thermal pulses in dielectric crystals. Lett. Appl. Eng. Sci., 1973, 1, 35-47). Thermodynamic restrictions on the constitutive equations and the domain of dependence are first derived. Then, in the unidimensional case, existence, uniqueness and asymptotic stability properties of solution are investigated. Finally, we also prove some dissipative properties of the system.