  We establish the long-time existence of large-data weak solutions to a system
of nonlinear partial differential equations. The system of interest governs the
motion of non-Newtonian fluids described by a simplified viscoelastic rate-type
model with a stress-diffusion term. The simplified model shares many
qualitative features with more complex viscoelastic rate-type models that are
frequently used in the modeling of fluids with complicated microstructure. As
such, the simplified model provides important preliminary insight into the
mathematical properties of these more complex and practically relevant models
of non-Newtonian fluids. The simplified model that is analyzed from the
mathematical perspective is shown to be thermodynamically consistent, and we
extensively comment on the interplay between the thermodynamical background of
the model and the mathematical analysis of the corresponding
initial-boundary-value problem.
