2022/06/30 by Barbara Kaltenbacher, Mostafa Meliani, Kaltenbacher, Barbara +3 · 1 citation
Computer Science · Engineering · Mathematics · #35L05 #35L72 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2206.15245
openalex publication_date 2022/06/30 · openalex created_date 2022/07/03 · openalex updated_date 2026/07/28
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.