Article Contents

2022, Volume 15, Issue 8: 1957-1985. Doi: 10.3934/dcdss.2022107

This issue Previous Article On the time decay for the MGT-type porosity problems Next Article Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

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

Marcelo Bongarti^1, and
Irena Lasiecka^2, ,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10777, Berlin, Germany
2.
Department of Mathematical Sciences, University of Memphis, 3725 Norriswood Ave, Memphis, TN 38152, United States, IBS, Polish Academy of Sciences, Warsaw

^* Corresponding author: Irena Lasiecka
^* Corresponding author: Irena Lasiecka

Received: December 2021

Revised: April 2022

Early access: May 2022

Published: August 2022

I. Lasiecka is supported by NSF grant DMS-1713506

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Abstract

Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.

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Mathematics Subject Classification: Primary: 35LXX; Secondary: 53C35.

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