  We study the Einstein-scalar field system with positive cosmological constant
and spherically symmetric characteristic initial data given on a truncated null
cone. We prove well-posedness, global existence and exponential decay in
(Bondi) time, for small data. From this, it follows that initial data close
enough to de Sitter data evolves to a causally geodesically complete spacetime
(with boundary), which approaches a region of de Sitter asymptotically at an
exponential rate; this is a non-linear stability result for de Sitter within
the class under consideration, as well as a realization of the cosmic no-hair
conjecture.
