  We establish global-in-time existence results for thermodynamically
consistent reaction-(cross-)diffusion systems coupled to an equation describing
heat transfer. Our main interest is to model species-dependent diffusivities,
while at the same time ensuring thermodynamic consistency. A key difficulty of
the non-isothermal case lies in the intrinsic presence of cross-diffusion type
phenomena like the Soret and the Dufour effect: due to the temperature/energy
dependence of the thermodynamic equilibria, a nonvanishing temperature gradient
may drive a concentration flux even in a situation with constant
concentrations; likewise, a nonvanishing concentration gradient may drive a
heat flux even in a case of spatially constant temperature. We use time
discretisation and regularisation techniques and derive a priori estimates
based on a suitable entropy and the associated entropy production. Renormalised
solutions are used in cases where non-integrable diffusion fluxes or reaction
terms appear.
