Inverse problems for a quasilinear strongly damped wave equation arising in nonlinear acoustics

We consider inverse problems for a Westervelt equation with a strong damping and a time-dependent potential $q$. We first prove that all boundary measurements, including the initial data, final data, and the lateral boundary measurements, uniquely determine $q$ and the nonlinear coefficient $\beta$. The proof is based on complex geometric optics construction and the approach proposed by Isakov. Further, by considering fundamental solutions supported in a half-space constructed by H\"ormander, we prove that with vanishing initial conditions the Dirichlet-to-Neumann map determines $q$ and $\beta$.