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2011, 2011(Special): 763-773. doi: 10.3934/proc.2011.2011.763

Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions

Barbara Kaltenbacher 1, and  Irena Lasiecka 2,

1. 

Institute of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria
2. 

Kerchof Hall , P. O. Box 400137, University of Virginia, Charlottesville, VA 22904-4137

Received  July 2010 Revised  January 2011 Published  October 2011

In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We prove local in time well-posedness as well as global existence and exponential decay for a slightly modi ed model. A key step of the proof is an appropriate extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by the strongly damped wave equation.
Keywords: nonlinear acoustics, local and global well- posedness, Kuznetsov's equation.
Mathematics Subject Classification: Primary: 35L72; Secondary: 35B40, 37N20.
Citation: Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
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