  Ultrasonic propagation through media with thermal and molecular relaxation
can be modeled by third-order in time nonlinear wave-like equations with
memory. This paper investigates the asymptotic behavior of a Cauchy problem for
such a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the
so-called critical case, which corresponds to propagation in inviscid fluids.
The memory has an exponentially fading character and type I, meaning that
involves only the acoustic velocity potential. A major challenge in the global
analysis is that the linearized equation's decay estimates are of
regularity-loss type. As a result, the classical energy methods fail to work
for the nonlinear problem. To overcome this difficulty, we construct
appropriate time-weighted norms, where weights can have negative exponents.
These problem-tailored norms create artificial damping terms that help control
the nonlinearity and the loss of derivatives, and ultimately allow us to
discover the model's asymptotic behavior.
