  The Jordan-Moore-Gibson-Thompson
(JMGT)\cite{christov_heat_2005,jordan_nonlinear_2008,straughan_heat_2014}
equation is a benchmark model describing propagation of nonlinear acoustic
waves in heterogeneous fluids at rest. This is a third-order (in time) dynamics
which accounts for a finite speed of propagation of heat signals (see
\cite{coulouvrat_equations_1992,crighton_model_1979,jordan_nonlinear_2008,jordan_second-sound_2014,kaltenbacher_jordan-moore-gibson-thompson_2019}).
In this paper, we study a boundary stabilization of linearized version (also
known as MGT-equation) in the {\it critical case}, configuration in which the
smallness of the diffusion effects leads to conservative dynamics
\cite{kaltenbacher_wellposedness_2011}. Through a single measurement in
{\it{feedback}} form made on a non-empty, relatively open portion of the
boundary under natural geometric conditions, we were able to obtain uniform
exponential stability results that are, in addition, uniform with respect to
the space-dependent viscoelasticity parameter which no longer needs to be
assumed positive and in fact can be degenerate and taken to be zero on the
whole domain. This result, of independent interest in the area of boundary
stabilization of MGT equations, provides a necessary first step for the study
of optimal boundary feedback control on {\it infinite horizon}
\cite{bucci_feedback_2019}.
