  In this paper we show local (and partially global) in time existence for the
Westervelt equation with several versions of nonlinear damping. This enables us
to prove well-posedness with spatially varying $L_\infty$-coefficients, which
includes the situation of interface coupling between linear and nonlinear
acoustics as well as between linear elasticity and nonlinear acoustics, as
relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
