  We consider an inverse problem for a Westervelt type nonlinear wave equation
with fractional damping. This equation arises in nonlinear acoustic imaging,
and we show the forward problem is locally well-posed. We prove that the smooth
coefficient of the nonlinearity can be uniquely determined, based on the
knowledge of the source-to-solution map and a priori knowledge of the
coefficient in an arbitrarily small subset of the domain. Our approach relies
on a second order linearization as well as the unique continuation property of
the spectral fractional Laplacian.
