Well-posedness and longtime 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 in Kaltenbacher and Shevchenko (Discrete Contin Dyn Syst 1000–1008, 2015), Shevchenko and Kaltenbacher (J Comput Phys 302:200–221, 2015). 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 ∞ 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.