Well-posedness and long-time behavior for the Westervelt equation with absorbing boundary conditions of order zero

Abstract: We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed by Kaltenbacher & Shevchenko. We apply the concept of maximal regularity of type $L_p$ to prove global well-posedness for small initial data. Moreover, we show that the solutions regularize instantaneously which means that they are $C<sup>\infty$ with respect to time $t$ as soon as $t&gt;0$. Finally, we show that each equilibrium is stable and each solution which starts sufficiently close to an equilibrium converges at an exponential rate to a possibly different equilibrium.