  We study coupled non-linear parabolic equations for a fluid described by a
material density and a temperature, both functions of space and time. In one
dimension, we find some stationary solutions corresponding to fixing the
temperature on the boundary, with no-escape boundary conditions for the
material. For the special case, where the temperature on the boundary is the
same at both ends, the linearised equations for small perturbations about a
stationary solution are derived; they are subject to the boundary conditions,
Dirichlet for the temperature and no-flow conditions for the material. The
spectrum of the generator L of time-evolution, regarded as an operator on the
Hilbert space of square-integrable functions on [0,1], is shown to be real,
discrete and non-positive, even though L is not self-adjoint. This result is
necessary for the stability of the stationary state, but might not be
sufficient. The problem lies in the fact that L is not a sectorial operator;
its numerical range is the whole of the complex plane.
