  Thermodynamics is continuously spreading in the engineering practice, which
is especially true for non-equilibrium models in continuum problems. Although
there are concepts and approaches beyond the classical knowledge, which are
known for decades, their mathematical properties and consequences of the
generalizations are less-known and are still of high interest in current
researches. Therefore, we found it essential to collect the most important and
still open mathematical questions related to different continuum thermodynamic
approaches. First, we start with the example of Classical Irreversible
Thermodynamics (CIT) in order to provide the basis for the more general and
complex frameworks, such as the Non-Equilibrium Thermodynamics with Internal
Variables (NET-IV) and Rational Extended Thermodynamics (RET). Here, we aim to
present that each approach has its specific problems, such as how the initial
and boundary conditions can be formulated, how the coefficients in the partial
differential equations are connected to each other, and how it affects the
appearance of nonlinearities. We present these properties and comparing the
approach of NET-IV and RET to each other from these points of view.
