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When high-frequency sound waves travel through media with anomalousdiffusion, such as biological tissues, their motion can be described bynonlinear wave equations of fractional higher order. These can be understood asnonlocal generalizations of the Jordan-Moore-Gibson-Thompson equations innonlinear acoustics. In this work, we relate them to the classical second-orderacoustic equations and, in this sense, justify them as their approximations forsmall relaxation times. To this end, we perform the singular limit analysis fora class of corresponding nonlocal wave models and determine their behavior asthe relaxation time tends to zero. We show that, depending on thenonlinearities and assumptions on the data, these models can be seen asapproximations of the Westervelt, Blackstock, or Kuznetsov wave equations. Theanalysis rests upon the uniform bounds for the solutions of the equations withfractional higher-order derivatives, obtained through a testing proceduretailored to the coercivity property of the involved (weakly) singular memorykernels.

Nonlinear acoustic equations of fractional higher order at the singular limit