Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation. (December 2021)

• Record Type: Journal Article
• Authors: Nikolić, Vanja; Said-Houari, Belkacem
• Is Part Of: Nonlinear analysis. Volume 62(2021)
• Publication Date: 2021-12
• Subjects: Nonlinear acoustics -- JMGT equation -- Memory of type I -- Global well-posedness -- Asymptotic behavior; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2021.103384
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation's decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model's asymptotic behavior.