The paper under review is concerned with the following system of nonlinear PDE’s in the cylinder \(\Omega \times (0, \tau)\): \[ \begin{aligned} \frac{\partial \rho_{i}}{\partial t}+ \text{div }J_{i} &= W_{i}\quad (i=1, \cdots, n), \tag{1}\\ \frac{\partial U}{\partial t}+ \text{div } J_{n+1}&= \sum^{n}_{k=1} F_{k} \cdot J_{k}+ W_{n+1},\tag{2}\\ -\Delta V&= \sum^{n}_{k=1 }e_{k} \rho_{k}+ C,\tag{3} \end{aligned} \] where \(\rho_{i}\) is the density, \(J_{i}\) the carrier flux density, \(e_{i}\) the particle charge \((i=1,\dots, n)\), \(U\) the internal energy density, \( J_{n+1}\) the heat flux, and \(C=C(x)\) a given function. Equations (1)–(3) model the evolution of a gas or fluid consisting of \(n\) components, in the bounded domain \( \Omega \subset \mathbb{R}^{d}\) \((d\geq 1)\). The following assumptions are made: \(\rho_{i}= \rho_{i}(u)\), \(\rho\) is strongly monotone, \(\rho= \bigtriangledown \chi\), \(F_{k} =-e_{k}\bigtriangledown V\), \(J_{i} =-\sum^{n+1}_{j=1} L_{ij}X_{j}\), where \[ u= \left\{ \frac{\mu_{1}}{T},\cdots,\frac{\mu_{1}}{T}, -\frac{1}{T} \right\}, \quad X_{i}= \bigtriangledown \left( \frac{\mu_{1}}{T}\right)+ e_{i} \frac{\bigtriangledown V}{T} \quad (i=1,\dots, n), \qquad X_{n+1}=- \bigtriangledown \left( \frac{1}{T}\right), \] where \(\mu_{i}\) is the chemical potential, and \(T\) is the temperature. Equations (1)–(3) are supplemented by mixed Dirichlet and Neumann conditions on \(u\) and \(V\) on \(\partial \Omega\), and the initial condition \(u( \cdot, 0)= u_{0}\) in \(\Omega\).
The authors introduce new dependent variables \(w_{i}= u_{i}- e_{i} u_{n+1}V\) \((i=1,\cdots, n)\), \(w_{n+1}=u_{n+1}\), which symmetrize the problem under consideration. For this problem the existence of a weak solution is proved (when \(d=3\) only Dirichlet conditions on the whole boundary \(\partial \Omega\) are considered). The method of proof is based on a semidiscretization of time, where the resulting elliptic problems are solved by using estimates on the discrete entropy function and the Leray-Schauder fixed point theorem.
In the last section the authors show that the solution tends to the thermal equilibrium state when \(\tau \rightarrow+ \infty\), if the boundary data are in thermal equilibrium.