2020/12/02 by Vanja Nikolić, Nikolić, Vanja, Belkacem Said‐Houari +1 · 1 citation
Computer Science · Engineering · Mathematics · #35G25 #35L75 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2012.01142
openalex publication_date 2020/12/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Ultrasonic propagation through media with thermal and molecular relaxation\ncan be modeled by third-order in time nonlinear wave-like equations with\nmemory. This paper investigates the asymptotic behavior of a Cauchy problem for\nsuch a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the\nso-called critical case, which corresponds to propagation in inviscid fluids.\nThe memory has an exponentially fading character and type I, meaning that\ninvolves only the acoustic velocity potential. A major challenge in the global\nanalysis is that the linearized equation's decay estimates are of\nregularity-loss type. As a result, the classical energy methods fail to work\nfor the nonlinear problem. To overcome this difficulty, we construct\nappropriate time-weighted norms, where weights can have negative exponents.\nThese problem-tailored norms create artificial damping terms that help control\nthe nonlinearity and the loss of derivatives, and ultimately allow us to\ndiscover the model's asymptotic behavior.\n