Stability of equilibria to the model for non-isothermal electrokinetics

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 dynamics of $N$-ionic species in a solvent derived. this paper we first reformulate PNPF ($4N+6$ equations) into an evolutional with $N+1$ equations, and define new total electrical charge. We then prove constant states are stable provided that they such perturbed systems around them dissipative. However, not all positive solutions corresponding characterize set equilibria $\mathcal{S}_{eq}$ whose elements satisfy conditions {(A1)} {(A2)}, it is nonempty. After then, stability these equilibria, thus global well-posedness near them.