Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity
Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity
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DOI：
10.3934/dcdss.2022020
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发表时间：
2021-07
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影响因子：
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通讯作者：
Marcelo Bongarti;I. Lasiecka;J. H. Rodrigues
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文献类型：
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作者：
Marcelo Bongarti;I. Lasiecka;J. H. Rodrigues
Jordan-Moore-Gibson-Thompson（JMGT）方程是近年来得到广泛研究的非线性声学模型。它是一个三阶（时间）半线性偏微分方程（PDE），具有预测超声波以有限速度传播的显着特征。这是由于被称为第二声音的热现象导致双曲线热波传播。在本文中，我们考虑在所谓的“临界”情况下，自由动力学是不稳定的问题。为了稳定，我们将使用仅在边界的一部分上支持的边界反馈控制。由于边界的其余部分不是“受控的”，并且施加的Neumann型边界条件不能满足Lopatinski条件，因此出现了边界稳定性背景下混合问题的几个典型数学问题。为了解决这些问题，将开发特殊的几何结构沿着与尖锐的跟踪估计。所施加的几何条件是由适合于对控制（从边界）在医学治疗（诸如碎石术、热疗、声化学或涉及高强度聚焦超声（HIFU）的任何其他过程）中涉及的声压的问题进行建模的几何结构激发的。
The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).