Billiard models of single particles moving freely in two-dimensional regions enclosed by hard walls, have long provided ideal toy models for the investigation of dynamical systems and chaos. Recently, billiards with (semi-)permeable walls and internal holes have been used to study open systems. Here we introduce a billiard model containing an internal region with partial absorption. The absorption does not change the trajectories, but instead reduces an intensity variable associated with each trajectory. The value of the intensity can be tracked as a function of the initial configuration and the number of reflections from the wall and depicted in intensity landscapes over the Poincar'e phase space. This is similar in spirit to escape time diagrams that are often considered in dynamical systems with holes. We analyse the resulting intensity landscapes for three different geometries; a circular, elliptic, and oval billiard, respectively, all with a centrally placed circular absorbing region. The intensity landscapes feature increasingly more complex structures, organised around the sets of points in phase space that intersect the absorbing region in a given iteration, which we study in some detail. On top of these, the intensity landscapes are enriched by effects arising from multiple absorption events for a given trajectory.