Limiting behavior of quasilinear wave equations with fractional-type dissipation

TL;DR

This paper studies the behavior of quasilinear wave equations with nonlocal fractional dissipation, proving well-posedness under weak assumptions and analyzing their limiting behavior as a key parameter approaches zero.

Contribution

It introduces a general framework for well-posedness of wave equations with weakly singular nonlocal kernels, including Abel and Mittag-Leffler types, and examines their asymptotic limits.

Findings

Proved well-posedness for equations with fractional and nonlocal kernels.

Analyzed the asymptotic behavior as the diffusivity parameter vanishes.

Connected the equations to their limiting models under various kernel classes.

Abstract

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