  In view of the properties of mesons in hot strongly interacting matter the
properties of the solutions of the truncated Dyson-Schwinger equation for the
quark propagator at finite temperatures within the rainbow-ladder approximation
are analysed in some detail. In Euclidean space within the Matsubara imaginary
time formalism the quark propagator is not longer a O(4) symmetric function and
possesses a discrete spectra of the fourth component of the momentum. This
makes the treatment of the Dyson-Schwinger and Bethe-Salpeter equations
conceptually different from the vacuum and technically much more involved. The
question whether the interaction kernel known from vacuum calculations can be
applied at finite temperatures remains still open. We find that, at low
temperatures, the model interaction with vacuum parameters provides a
reasonable description of the quark propagator, while at temperatures higher
than a certain critical value $T_c$ the interaction requires stringent
modifications. The general properties of the quark propagator at finite
temperatures can be inferred from lattice QCD calculations. We argue that, to
achieve a reasonable agreement of the model calculations with that from lattice
QCD, the kernel is to be modified in such a way as to screen the infra-red part
of the interaction at temperatures larger than $T_c$. For this, we analyse the
solutions of the truncated Dyson-Schwinger equation with existing interaction
kernels in a large temperature range with particular attention on high
temperatures in order to find hints to an adequate temperature dependence of
the interaction kernel to be further implemented in to the Bethe-Salpeter
equation for mesons. This will allow to investigate the possible in medium
modifications of the meson properties as well as the conditions of quark
deconfinement in hot matter.
