The authors establish existence and uniqueness of a solution for isothermal non-Newtonian bipolar compressible fluids for which the stress tensor can be expressed in the form of potentials depending on the rate of the strain tensor and on its spatial partial derivatives. They also study the stability of the rest state of bipolar isothermal fluids by defining a positive energy functional for perturbations which can be controlled by its value at the initial time. To achieve this, a general procedure due to \textit{A. Novotny} and \textit{M. Padula} [Int. J. Non-Linear Mech. 28, No. 1, 117-130 (1993; Zbl 0774.76009)] is used. It has been established that the perturbations of velocity and density tend to zero in appropriate norms along a suitably chosen time sequence. Unconditional asymptotic stability of the rest state for regular potential forces also has been obtained. Formulations of related open problems conclude the paper.