  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 $C^\infty$ with respect to time $t$ as soon as $t>0$. 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.
