  Recently, energetic variational approach was employed to derive models for
non-isothermal electrokinetics by Liu et. al \cite{Liu-Wu-Liu-CMS2018}. In
particular, the Poisson-Nernst-Planck-Fourier (PNPF) system for the dynamics of
$N$-ionic species in a solvent was derived. In this paper we first reformulate
PNPF ($4N+6$ equations) into an evolutional system with $N+1$ equations, and
define a new total electrical charge. We then prove the constant states are
stable provided that they are such that the perturbed systems around them are
dissipative. However, not all positive constant solutions of PNPF are such that
the corresponding perturbed systems are dissipative. We characterize a set of
equilibria $\mathcal{S}_{eq}$ whose elements satisfy the conditions {(A1)} and
{(A2)}, and prove it is nonempty. After then, we prove the stability of these
equilibria, thus the global well-posedness of PNPF near them.
