A steady state analytical solution of the heat flux equation in the 13-moment approximation is obtained, where temperature gradient, regimes changing from subsonic to supersonic flow situations, and effects of drift velocity between species, say diffusion-thermal effects, are taken into account. From such a general solution, validity conditions of Fourier's law are examined. A solution for a collisionless regime based upon the above solution is achieved. Then, the high-latitude topside ionosphere with a mixture of the ionized constituents O+ and H+ and neutrals is studied. Heat flux is derived for the two ionized constituents in a regime where the gradient of the aligned velocity component is less important than collision frequencies. Heat fluxes of O+ and H+ ions in this situation are further simplified considering that H+ ion concentration is smaller than O+ ion concentration. With the help of the European incoherent scatter (EISCAT)-VHF data, O+ and H+ heat fluxes in the high-latitude topside ionosphere are calculated with the above solutions. Results are compared to those obtained with Fourier's law. We find that Fourier's law underestimates significantly heat fluxes in this case, especially for H+. Thermal conduction is calculated, both with our general solution and with the Fourier's law. For the H+ ion thermal conduction, Fourier's law may lead to an inverse variation with altitude when compared to our solution, which gives an inverse contribution to the H+ ion energy balance and to the H+ temprature. H+ ion heat flux is upward not only in the supersonic regime shown by DE 1 data but also in the subsonic regime at lower altitude in the high-latitude topside ionosphere, as shown by EISCAT data. ¿ American Geophysical Union 1993