  We propose a system of two coupled parabolic equations that have degenerate
diffusion constants depending on the energy-like variable. The dissipation of
the velocity-like variable is fed as a source term into the energy equation
leading to conservation of the total energy. Because of the degeneracies there
are solutions with nontrivial support that may grow in time like in the porous
medium equation, which is contained in our system as a special case. The motion
of the free boundary may be driven by either self-diffusion of the energy-like
variable or by dissipation of the velocity-like variable. We discuss the
cross-over of these two phenomena in terms of the associated planar traveling
waves. Moreover, we provide existence of suitably defined weak and very weak
solutions. After providing a thermodynamically motivated gradient structure we
also establish convergence into steady state for bounded domains.
