Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Abstract

Boundary feedback stabilization of a critical, nonlinear Jordan-Moore-Gibson-Thompson (JMGT) equation is considered. JMGT arises in modeling of acoustic waves involved in medical/engineering treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU). It is a well-established and recently widely studied model for nonlinear acoustics (NLA): a third-order (in time) semilinear Partial Differential Equation (PDE) with the distinctive feature of predicting the propagation of ultrasound waves at finite speed due to heat phenomenon know as second sound which leads to the hyperbolic character of heat propagation. In practice, the JMGT dynamics is largely used for modeling the evolution of the acoustic velocity and, most importantly, the acoustic pressure as sound waves propagate through certain media. In this work, critical refers to (usual) case where mediadamping effects are non-existent or non-measurable and therefore cannot be relied upon for stabilization purposes.In this paper the issue of boundary stabilizability of originally unstable (JMGT) equation is resolved. Motivated by modeling aspects in HIFU technology, boundary feedback is supported only on a portion of the boundary, while the remaining part of the boundary is left free (available to control actions) . Since the boundary conditions imposed on the "free" part of the boundary fail to satisfy Lopatinski condition (unlike Dirichlet boundary conditions), the analysis of uniform stabilization from the boundary becomes very subtle and requires careful geometric considerations.