Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

TL;DR

This paper investigates the boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity, relevant for controlling ultrasound waves in medical applications like HIFU.

Contribution

It introduces a stabilization approach for the linear MGT equation with specific boundary conditions and viscoelastic effects, advancing control methods for nonlinear acoustics models.

Findings

Established boundary stabilization results for the linear MGT equation.

Analyzed effects of partially absorbing boundary data and degenerate viscoelasticity.

Potential applications in medical ultrasound treatments like lithotripsy and thermotherapy.

Abstract

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) model with the distinctive feature of predicting the propagation of ultrasound waves at \textit{finite} speed due to heat phenomenon know as \textit{second sound} which leads to the hyperbolic character of heat propagation. In this paper, we consider the problem of stabilizability of the linear (known as) MGT--equation. We consider a special geometry that is suitable for studying the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU).