  When high-frequency sound waves travel through media with anomalous
diffusion, such as biological tissues, their motion can be described by
nonlinear wave equations of fractional higher order. These can be understood as
nonlocal generalizations of the Jordan-Moore-Gibson-Thompson equations in
nonlinear acoustics. In this work, we relate them to the classical second-order
acoustic equations and, in this sense, justify them as their approximations for
small relaxation times. To this end, we perform the singular limit analysis for
a class of corresponding nonlocal wave models and determine their behavior as
the relaxation time tends to zero. We show that, depending on the
nonlinearities and assumptions on the data, these models can be seen as
approximations of the Westervelt, Blackstock, or Kuznetsov wave equations. The
analysis rests upon the uniform bounds for the solutions of the equations with
fractional higher-order derivatives, obtained through a testing procedure
tailored to the coercivity property of the involved (weakly) singular memory
kernels.
