The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces $ \mathbb{R}^{N} $. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [8] and [26], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the $ L^{2} $-norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.

中文翻译：

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带记忆的摩尔-吉布森-汤普森方程的衰减率

本文的主要目标是研究具有记忆项的 Moore-Gibson-Thompson 方程 (MGT) 在整个空间 $\mathbb{R}^{N} $ 中解的存在性和稳定性。MGT 方程源于将高频超声波建模为众所周知的库兹涅佐夫方程的替代模型。首先，按照 [8] 和 [26]，我们表明在对系统系数的适当假设下问题是适定的。然后，我们利用傅立叶空间中的能量方法建立了一些李雅普诺夫泛函。这些泛函使我们能够获得对解决方案的傅立叶图像的控制估计。傅立叶图像的这些估计值与一些积分不等式一起导致解的 $L^{2}$-范数的衰减率。我们在这里使用两种类型的记忆词：Ⅰ型记忆词和Ⅲ型记忆词。在这两种类型中都可以获得衰减率。更准确地说，解的衰减率取决于内存内核的指数或多项式衰减。更重要的是，我们在两种情况下都显示了解决方案的稳定性：参数的亚临界范围和临界范围。