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In this work, we investigate a class of quasilinear wave equations ofWestervelt type with, in general, nonlocal-in-time dissipation. They arise asmodels of nonlinear sound propagation through complex media with anomalousdiffusion of Gurtin--Pipkin type. Aiming at minimal assumptions on the involvedmemory kernels -- which we allow to be weakly singular -- we prove thewell-posedness of such wave equations in a general theoretical framework. Inparticular, the Abel fractional kernels, as well as Mittag-Leffler-typekernels, are covered by our results. The analysis is carried out uniformly withrespect to the small involved parameter on which the kernels depend and whichcan be physically interpreted as the sound diffusivity or the thermalrelaxation time. We then analyze the behavior of solutions as this parametervanishes, and in this way relate the equations to their limiting counterparts.To establish the limiting problems, we distinguish among different classes ofkernels and analyze and discuss all ensuing cases.

Limiting behavior of quasilinear wave equations with fractional-type dissipation