2023/08/21 by Barbara Kaltenbacher, Mostafa Meliani, Kaltenbacher, B. +3 · 1 citation
Engineering · Mathematics · #35L05 #35L72 #35R11 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.10580
openalex publication_date 2023/08/21 · openalex created_date 2023/08/23 · openalex updated_date 2026/08/01
In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin--Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag-Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.