Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

Stefan Meyer; Mathias Wilke

doi:10.3934/eect.2013.2.365

Article Contents

2013, Volume 2, Issue 2: 365-378. Doi: 10.3934/eect.2013.2.365

This issue Previous Article Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement Next Article Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system

Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

Stefan Meyer^1, and
Mathias Wilke^1,

1.
Martin Luther University Halle-Wittenberg, NWF II - Institute of Mathematics, D - 06099 Halle (Saale)

Received: November 2012

Revised: February 2013

Published: June 2013

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Abstract

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.

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Mathematics Subject Classification: Primary: 35K59; Secondary: 35K51, 35Q35, 35B30, 35B35, 35B40, 35B45, 35B65.

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