  The Goldstein-Taylor equations can be thought of as a simplified version of a
BGK system, where the velocity variable is constricted to a discrete set of
values. It is intimately related to turbulent fluid motion and the
telegrapher's equation. A detailed understanding of the large time behaviour of
the solutions to these equations has been mostly achieved in the case where the
relaxation function, measuring the intensity of the relaxation towards equally
distributed velocity densities, is constant. The goal of the presented work is
to provide a general method to tackle the question of convergence to
equilibrium when the relaxation function is not constant, and to do so as
quantitatively as possible. In contrast to the usual modal decomposition of the
equations, which is natural when the relaxation function is constant, we define
a new Lyapunov functional of pseudodifferential nature, one that is motivated
by the modal analysis in the constant case, that is able to deal with full
spatial dependency of the relaxation function. The approach we develop is
robust enough that one can apply it to multi-velocity Goldstein-Taylor models,
and achieve explicit rates of convergence. The convergence rate we find,
however, is not optimal, as we show by comparing our result to the that found
in the work of Bernard and Salvarani from 2013.
