The paper deals with the coupled system \[ \psi_t=-(1-\alpha)\psi-\sigma \theta_x +\alpha\psi_{xx}, \] \[ \theta_t=-(1-\alpha)\theta+\nu \psi_x +2\psi \theta_{x}+\alpha\theta_{xx}, \] where \(\alpha\) and \(\nu\) are positive numbers. This set of equations singles out the essential mechanism of nonlinear interaction between ellipticity and dissipation. It is proved that the considered system is linearly stable if and only if \[ \nu< 4\alpha (1- \alpha). \] Moreover, if the initial data is small enough, the optimal decay rate has been obtained. Theorem 5 of the paper is in some sense closely related to Theorem 15.5.1 from the book of the reviewer [Stability of finite and infinite dimensional systems, Kluwer Academic Publishers, Boston-Dordrecht-London (1998; Zbl 0916.93002)].