Report
Limiting behavior of quasilinear wave equations with fractional-type dissipation
| العنوان: | Limiting behavior of quasilinear wave equations with fractional-type dissipation |
|---|---|
| المؤلفون: | Kaltenbacher, Barbara, Meliani, Mostafa, Nikolić, Vanja |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35L05, 35L72 |
| الوصف: | 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. Comment: Earlier version of this manuscript has been split into 2 parts to improve the readability. This is the first part focusing on a nonlocal Westervelt equation. A separate upload now contains a revised analysis of Kuznetsov's and Blackstock's equations |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2206.15245 |
| رقم الانضمام: | edsarx.2206.15245 |
| قاعدة البيانات: | arXiv |
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