September  2009, 2(3): 503-523. doi: 10.3934/dcdss.2009.2.503

Global existence and exponential decay rates for the Westervelt equation

1. 

Department of Mathematics, University of Stuttgart, 70569 Stuttgart, Germany

2. 

University of Virginia, Department of Mathematics, Charlottesville, VA 22901

Received  October 2008 Revised  December 2008 Published  June 2009

We consider the Westervelt equation which models propagation of sound in a fluid medium. This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy. The PDE model consists of the second order in time evolution which is both quasilinear and degenerate. Degeneracy depends on the fluctuations of the acoustic pressure.
    Our main results are : (1) global well-posedness, (2) exponential decay rates for the energy function corresponding to both weak and strong solutions. The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits analyticity properties of the underlying linearised semigroup, (ii) exploitation of decay rates associated with the dissipative mechanism along with barrier's method leading to global wellposedness. The obtained result holds for all times, provided that the initial data are taken from a suitably small ball characterized by the parameters of the equation.
Citation: Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503
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