  We study a time-reversed hyperbolic heat conduction problem based upon the
Maxwell--Cattaneo model of non-Fourier heat law. This heat and mass diffusion
problem is a hyperbolic type equation for thermodynamics systems with thermal
memory or with finite time-delayed heat flux, where the Fourier or Fick law is
proven to be unsuccessful with experimental data. In this work, we show that
our recent variational quasi-reversibility method for the classical
time-reversed heat conduction problem, which obeys the Fourier or Fick law, can
be adapted to cope with this hyperbolic scenario. We establish a generic
regularization scheme in the sense that we perturb both spatial operators
involved in the PDE. Driven by a Carleman weight function, we exploit the
natural energy method to prove the well-posedness of this regularized scheme.
Moreover, we prove the H\"older rate of convergence in the mixed $L^2$--$H^1$
spaces.
