  In ultrasonics, nonlocal quasilinear wave equations arise when taking into
account a class of heat flux laws of Gurtin--Pipkin type within the system of
governing equations of sound motion. The present study extends previous work by
the authors to incorporate nonlocal acoustic wave equations with quadratic
gradient nonlinearities which require a new approach in the energy analysis.
More precisely, we investigate the Kuznetsov and Blackstock equations with
dissipation of fractional type and identify a minimal set of assumptions on the
memory kernel needed for each equation. In particular, we discuss the
physically relevant examples of Abel and Mittag-Leffler kernels. We perform the
well-posedness analysis uniformly with respect to a small parameter on which
the kernels depend and which can be interpreted as the sound diffusivity or the
thermal relaxation time. We then analyze the limiting behavior of solutions
with respect to this parameter, and how it is influenced by the specific class
of memory kernels at hand. Through such a limiting study, we relate the
considered nonlocal quasilinear equations to their limiting counterparts and
establish the convergence rates of the respective solutions in the energy norm.
