The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter \(\tau >0\), the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when \(\tau \rightarrow 0 \). It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when \(\tau \rightarrow 0\) has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

（时间上的三阶）JMGT方程[Jordan（J Acoust Soc Am 124（4）：2491-249，2008）和Cattaneo（C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3：83-101，1948）]是非线性（准线性）偏微分方程（PDE）模型，用于描述声音在声学介质中的非线性传播。重要的特征是该模型避免了与经典二阶时间方程（称为Westervelt方程）相关的无限传播速度悖论。用麦克斯韦-卡塔尼定律代替傅立叶定律，产生了由小参数\（\ tau> 0 \）缩放的三阶时间导数，后者代表热弛豫时间参数，并且是发生动力学的介质所固有的。在本文中，我们提供了\（\ tau \ rightarrow 0 \）时三阶模型的渐近分析。结果表明，相应的解在一个强相空间拓扑中收敛到一个极限，该极限是韦斯特维尔特方程的解。另外，为显示高阶规则性的解决方案提供了收敛速度。这解决了[20]中提出的一个开放性问题，其中已经研究了一个相关的JMGT方程，并且在建立\（\ tau \ rightarrow 0 \）时解的弱星形收敛。因此，我们的主要贡献在于无限时间范围内的强收敛性，以及相关的收敛速率在有限时间范围内有效。解锁难度的关键在于严格控制和传播初始数据的“小巧”，以便在三个不同的拓扑级别进行估计。收敛速度允许人们估计信号到达目标所需的弛豫时间。研究这类问题的兴趣是由工程和医学领域中出现的大量应用激发的。