Nonlinear acoustic equations of fractional higher order at the singular limit

Abstract: 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.