  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 $\beta$.
