  The main purpose of this article is the study of an inverse problem for
nonlocal porous medium equations (NPMEs) with a linear absorption term. More
concretely, we show that under certain assumptions on the time-independent
coefficients $\rho,q$ and the time-independent kernel $K$ of the nonlocal
operator $L_K$, the (partial) Dirichlet-to-Neumann map uniquely determines the
three quantities $(\rho,K,q)$ in the nonlocal porous medium equation $\rho
\partial_tu+L_K(u^m)+qu=0$, where $m>1$. In the first part of this work we
adapt the Galerkin method to prove existence and uniqueness of nonnegative,
bounded solutions to the homogenoeus NPME with regular initial and exterior
conditions. Additionally, a comparison principle for solutions of the NPME is
proved, whenever they can be approximated by sufficiently regular functions
like the one constructed for the homogeneous NPME. These results are then used
in the second part to prove the unique determination of the coefficients
$(\rho,K,q)$ in the inverse problem. Finally, we show that the assumptions on
the nonlocal operator $L_K$ in our main theorem are satisfied by the fractional
conductivity operator $\mathcal{L}_{\gamma}$, whose kernel is
$\gamma^{1/2}(x)\gamma^{1/2}(y)/|x-y|^{n+2s}$ up to a normalization constant.
